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Finitely generated group

In abstract algebra, a finitely generated group is a group G that admits a finite generating set S \subseteq G, meaning every element of G can be expressed as a finite product involving elements of S and their inverses under the group operation.^[1] This concept is central to group theory, as it captures groups that can be "built" from a small number of basic elements, contrasting with infinitely generated groups like the rational numbers under addition.^[2] All finite groups are finitely generated, since the group itself serves as a finite generating set, but the converse does not hold: there exist infinite finitely generated groups, such as the infinite cyclic group \mathbb{Z} generated by $1, free groups on a finite number of generators.^[1] Subgroups and quotients of finitely generated groups are also finitely generated, though not all subgroups of a finitely generated group need be (e.g., the commutator subgroup of the free group on two generators is free on countably infinitely many generators).^[1]^[3] A key property is that finitely generated groups have only finitely many subgroups of any given finite index, which has implications for their structure and classification.^[4] For abelian finitely generated groups, the fundamental theorem provides a complete classification: every such group is isomorphic to a direct sum of a finite number of cyclic groups, either infinite (\mathbb{Z}) or finite (\mathbb{Z}/n\mathbb{Z}).^[1] More generally, finitely generated groups underpin areas like geometric group theory, where their Cayley graphs (with respect to a finite generating set) encode metric and combinatorial properties, and computational group theory, where challenges like the word problem highlight undecidability for certain presentations.^[5] The study of finitely generated groups also intersects with topology and number theory, as seen in braid groups and profinite completions.^[5]

Fundamentals

Definition

A group G is finitely generated if there exists a finite subset S \subseteq G such that every element of G can be expressed as a finite product of elements from S and their inverses.^[6] The subgroup generated by S, denoted \langle S \rangle, is the smallest subgroup of G containing S, and G is finitely generated precisely when G = \langle S \rangle for some finite S.^[7] In this context, the size of the generating set |S| is finite, distinguishing finitely generated groups from those requiring infinitely many generators, such as the direct sum of countably many copies of the cyclic group of order 2.^[6] Key results on finitely generated subgroups of free groups were established in the early 20th century by Jakob Nielsen and Otto Schreier. Nielsen's 1921 work established that finitely generated subgroups of free groups are themselves free, laying groundwork for understanding generation in group theory.^[8] Schreier later extended these results to arbitrary subgroups, further solidifying the framework for finitely generated structures.^[9] Unlike finite groups, which are always finitely generated by their entire element set, finitely generated groups may be infinite; for instance, the additive group of integers \mathbb{Z} is generated by the singleton set \{1\}, as every integer is a finite sum of 1's and -1's.^[7] The trivial group, generated by the empty set, is a boundary case of finite generation, though typical discussions emphasize non-trivial finite generating sets for non-trivial groups.^[6]

Presentations

A presentation of a group G is a formal description given by a set S of generators and a set R of relations, denoted \langle S \mid R \rangle, where elements of R are words in the generators S and their inverses that are required to equal the identity element in G.^[10] This construction arises from the free group on S, with relations imposed by setting the words in R to the identity.^[10] Von Dyck's theorem provides the foundational link between such presentations and actual groups: if G is generated by a set S and satisfies the relations R (meaning each word in R equals the identity in G), then G is isomorphic to the quotient of the free group on S by the normal closure of R.^[11] This theorem, established in 1882, formalizes how presentations capture group structure through generators and imposed relations.^[11] When both the generating set S and the set of relations R are finite, the presentation is called finite, and the resulting group is finitely presented—a proper subclass of the finitely generated groups.^[12] Finitely presented groups allow compact algebraic descriptions but may exhibit complex behavior, such as undecidability in certain membership problems.^[12] A classic example is the cyclic group \mathbb{Z}/n\mathbb{Z}, presented as \langle a \mid a^n = 1 \rangle, where a generates the group and the single relation enforces the order n.^[10] Tietze transformations offer systematic ways to modify a presentation while preserving the isomorphism class of the group, including adding or deleting generators and relations under specific conditions, such as introducing a new generator defined by an existing word or eliminating a generator that appears in only one relation.^[13] These operations, introduced in 1908, enable equivalence checks between different presentations of the same group.^[13]

Properties

Subgroups

Subgroups of finitely generated groups exhibit diverse structural properties, ranging from being finitely generated themselves to displaying significant distortion or infinite generation. A key result concerning free groups, which are archetypal finitely generated groups, is the Nielsen-Schreier theorem, which asserts that every subgroup of a free group is itself free. Specifically, if G is a free group of finite rank r and H is a finite-index subgroup with [G : H] = n, then the rank of H is given by the formula \operatorname{rank}(H) = (r-1)n + 1. This theorem highlights the robustness of free groups under subgroup formation, ensuring that all subgroups, including those of infinite index, remain free, though potentially of infinite rank. Not all subgroups of finitely generated groups are finitely generated, providing a stark contrast to the free group case. For instance, the free group of rank 2 contains the commutator subgroup, which is free of countable infinite rank and thus infinitely generated. Another example arises in the Baumslag-Solitar group BS(1,2) = \langle a, t \mid t a t^{-1} = a^2 \rangle, which is finitely generated but possesses the normal closure of \langle a \rangle isomorphic to the additive group of dyadic rationals \mathbb{Z}[1/2] and is not finitely generated.^[14] Similarly, the Baumslag-Solitar group BS(2,3) = \langle a, t \mid t a^2 t^{-1} = a^3 \rangle has infinitely generated subgroups, such as the normal closure of \langle a \rangle. In contrast, for the abelian finitely generated group \mathbb{Z}^2, all subgroups are finitely generated, as guaranteed by the fundamental theorem of finitely generated abelian groups. In the context of hyperbolic groups, a class of finitely generated groups introduced by Gromov with negatively curved Cayley graphs, the notion of subgroup distortion becomes particularly relevant. A subgroup H of a hyperbolic group G is quasi-convex if there exists a constant \delta > 0 such that every geodesic in the Cayley graph of G between two points in H stays within \delta of H. Quasi-convex subgroups are undistorted, meaning the inclusion H \hookrightarrow G is a quasi-isometric embedding with respect to word metrics, and such subgroups are themselves hyperbolic. This property ensures that quasi-convex subgroups preserve the hyperbolic geometry of the ambient group without pathological stretching. The growth of subgroups, particularly the enumeration of those of finite index, provides quantitative insight into the subgroup structure of finitely generated groups. For a finitely generated group G, let a_n(G) denote the number of subgroups of index n; the subgroup growth function is then \sum_{k=1}^N a_k(G), which measures the total number of finite-index subgroups up to index N.^[15] This growth can be polynomial, exponential, or intermediate, and for groups like free groups, a_n(G) is explicitly computable and grows exponentially.^[15] In linear groups, subgroup growth is linked to representation theory, where the number of index-n subgroups relates to the dimensions of irreducible representations.^[15] Exotic examples, such as Tarski monster groups—finitely generated infinite groups where every proper nontrivial subgroup is cyclic of prime order p—demonstrate extreme restrictions on subgroup diversity, with all proper subgroups being finite and thus finitely generated, though the groups themselves are highly non-abelian and simple.

Quotients and Extensions

In group theory, quotients of finitely generated groups inherit the property of finite generation. Specifically, if G is a finitely generated group and N is a normal subgroup of G, then the quotient group G/N is also finitely generated. This follows from the fact that if G is generated by a finite set S, then the images of the elements of S under the canonical projection generate G/N.^[16] The correspondence theorem establishes a bijective correspondence between the normal subgroups of G containing N and the subgroups of the quotient G/N. Under this correspondence, the quotient map induces an isomorphism between the lattice of such normal subgroups and the lattice of subgroups of G/N, facilitating the analysis of normal subgroup structures through their quotients.^[16] A representative example of quotient construction is the fundamental group of a closed orientable surface of genus g \geq 2, which is obtained as the quotient of the free group F_{2g} on $2g generators by the normal subgroup generated by the single relator [a_1, b_1] \cdots [a_g, b_g], where [x, y] = x y x^{-1} y^{-1}. This yields a finitely presented group that is finitely generated and captures the topology of the surface.^[17] Group extensions are described by short exact sequences of the form $1 \to N \to G \to Q \to 1, where N is a normal subgroup of G (the kernel of the surjection G \to Q). If both N and Q are finitely generated, then G is finitely generated: choose a finite set of lifts in G of a finite generating set for Q, and adjoin a finite generating set for N; these together generate G. However, finite generation does not hold in the converse direction: G and Q may be finitely generated while N is not. The Baumslag-Solitar group BS(1,2) = \langle a, t \mid t a t^{-1} = a^2 \rangle provides such an example, as it is finitely generated by \{a, t\}, the quotient BS(1,2)/\langle a \rangle^{BS(1,2)} \cong \mathbb{Z} (generated by the image of t) is finitely generated, but the normal closure \langle a \rangle^{BS(1,2)} is isomorphic to the additive group of dyadic rationals \mathbb{Z}[1/2], which is not finitely generated.^[18] Central extensions form a subclass of group extensions in which the normal subgroup N lies in the center of G. The general finite generation preservation from the short exact sequence case applies here as well, so if N and Q are finitely generated, the central extension G is finitely generated. A canonical example is the discrete Heisenberg group over \mathbb{Z}, defined as the central extension $1 \to \mathbb{Z} \to H \to \mathbb{Z}^2 \to 1, where H consists of $3 \times 3 upper-triangular matrices with ones on the diagonal and integer off-diagonal entries, under matrix multiplication. This group is finitely generated (by the standard basis matrices corresponding to the off-diagonal positions) and nilpotent of class 2, illustrating how central extensions can yield non-abelian structures from abelian factors.^[19]

Abelian Case

Structure Theorem

The fundamental theorem of finitely generated abelian groups classifies such groups up to isomorphism. Specifically, every finitely generated abelian group G is isomorphic to a direct sum of the form \mathbb{Z}^r \oplus \bigoplus_{i=1}^k \mathbb{Z}/n_i\mathbb{Z}, where r \geq 0 is an integer known as the rank of G (the torsion-free rank, equal to the dimension of G \otimes_{\mathbb{Z}} \mathbb{Q} as a \mathbb{Q}-vector space), the n_i are positive integers satisfying n_i divides n_{i+1} for each i = 1, \dots, k-1, and k \geq 0 (with the understanding that the torsion sum is trivial if k=0).^[20] This decomposition separates G into its free abelian part \mathbb{Z}^r and its torsion subgroup T(G) = \bigoplus_{i=1}^k \mathbb{Z}/n_i\mathbb{Z}, where T(G) consists precisely of the elements of finite order in G, and is itself a finite abelian group.^[20] There are two standard canonical forms for this classification: the invariant factor decomposition (as stated above, with the n_i being the invariant factors) and the elementary divisor decomposition, where G \cong \mathbb{Z}^r \oplus \bigoplus_{j} \mathbb{Z}/p_j^{e_j}\mathbb{Z} for primes p_j and exponents e_j \geq 1. Both forms are unique up to isomorphism and the ordering of summands: the rank r is unique, the invariant factors n_1, \dots, n_k (with n_i > 1) are unique, and the elementary divisors (the prime power orders) are unique up to permutation.^[21] The two forms are related by factoring the invariant factors into prime powers; for instance, if n_3 = 12 = 2^2 \cdot 3, then \mathbb{Z}/12\mathbb{Z} \cong \mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z}.^[21] To prove the theorem, represent G via a presentation: since G is finitely generated abelian, there exist integers m \geq 0 and a surjective homomorphism \phi: \mathbb{Z}^m \twoheadrightarrow G with finitely generated kernel K = \operatorname{im} A, where A is an m \times m integer matrix whose columns generate K (after choosing bases).^[22] As \mathbb{Z} is a principal ideal domain (PID), the structure theorem for finitely generated modules over a PID applies: there exist invertible integer matrices P and Q (with \det P, \det Q = \pm 1) such that PAQ = D, where D = \operatorname{diag}(d_1, \dots, d_s, 0, \dots, 0) is the Smith normal form of A with s \leq m, the d_i > 0 are integers satisfying d_i divides d_{i+1} for each i, and trailing zeros account for the kernel's rank deficiency.^[23] The cokernel of D is then G \cong \mathbb{Z}^m / \operatorname{im} D \cong \bigoplus_{i=1}^s \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{m-s}, yielding the invariant factor form with r = m-s and n_i = d_i (discarding any d_i = 1).^[22] Uniqueness follows from the uniqueness of the Smith normal form and the fact that isomorphic presentations yield equivalent forms.^[21] The elementary divisor form is obtained by further decomposing each \mathbb{Z}/d_i\mathbb{Z} via the Chinese Remainder Theorem into primary components.^[20] For example, consider G \cong \mathbb{Z}^2 \oplus \mathbb{Z}/6\mathbb{Z}. The torsion subgroup is \mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} by the Chinese Remainder Theorem (since $6=2 \cdot 3 and 2, 3 are coprime), so G \cong \mathbb{Z}^2 \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} in elementary divisor form, or equivalently \mathbb{Z}^2 \oplus \mathbb{Z}/6\mathbb{Z} in invariant factor form (with single invariant factor 6).^[21] This illustrates how the decompositions capture the structure without altering the isomorphism type.

Invariants and Classification

Finitely generated abelian groups are classified up to isomorphism by a set of computable invariants derived from their decomposition into a free part and a torsion part. The rank r, also known as the Betti number, measures the dimension of the free abelian component \mathbb{Z}^r, and is determined by the maximal number of linearly independent elements over \mathbb{Z} in the group. This invariant captures the "free" rank and remains unchanged under isomorphism.^[24]^[25] The torsion part is characterized by torsion coefficients, which can be expressed either as invariant factors d_1, d_2, \dots, d_t where d_i divides d_{i+1} for each i, or as elementary divisors consisting of prime power orders. These coefficients satisfy gcd conditions: for invariant factors, each d_i divides the next, ensuring the group is isomorphic to \mathbb{Z}_{d_1} \oplus \mathbb{Z}_{d_2} \oplus \dots \oplus \mathbb{Z}_{d_t} \oplus \mathbb{Z}^r ; for elementary divisors, they group into primary components without such divisibility between different primes. These invariants uniquely determine the torsion subgroup up to isomorphism.^[26] In algebraic topology, finitely generated abelian groups frequently arise as homology groups of manifolds, where the Betti number r corresponds to the rank of the free part in H_k(M; \mathbb{Z}), and the torsion coefficients describe the finite cyclic summands. For instance, the first homology group of a torus is H_1(T^2; \mathbb{Z}) \cong \mathbb{Z}^2, with rank 2 and no torsion. To compute these invariants from a presentation of the group, algorithms based on normal forms of integer matrices are employed. Given a presentation matrix defining the relations, the Hermite normal form provides a row-reduced echelon form over \mathbb{Z}, from which the rank r is obtained as the number of zero rows, and the invariant factors are read from the diagonal entries after accounting for unimodular transformations. Alternatively, the Smith normal form diagonalizes the matrix with elementary divisors directly appearing as the non-trivial diagonal elements, enabling efficient decomposition even for large matrices via randomized or parallel methods. These computations run in polynomial time relative to the matrix size and entry magnitudes.^[26]^[27] A concrete example is the classification of abelian groups of order p^n for a prime p, which are torsion groups with no free part (rank 0). Such groups decompose uniquely as direct sums of cyclic p-groups: \mathbb{Z}/p^{c_1}\mathbb{Z} \oplus \mathbb{Z}/p^{c_2}\mathbb{Z} \oplus \dots \oplus \mathbb{Z}/p^{c_s}\mathbb{Z}, where c_1 \leq c_2 \leq \dots \leq c_s are positive integers summing to n in their exponents (i.e., c_1 + \dots + c_s = n). The number of non-isomorphic groups equals the number of partitions of n, with elementary divisors given by the prime powers p^{c_i}. For n=3, the possibilities are \mathbb{Z}/p^3\mathbb{Z}, \mathbb{Z}/p^2\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}, and \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}.^[28]

Examples

Cyclic and Free Groups

A cyclic group is the simplest example of a finitely generated group, requiring only one generator. The infinite cyclic group, isomorphic to the additive group of integers \mathbb{Z}, is generated by the element 1 and consists of all integer multiples k \cdot 1 for k \in \mathbb{Z}.^[29] It admits the presentation \langle a \mid \rangle, imposing no relations beyond the group axioms.^[30] Finite cyclic groups, denoted \mathbb{Z}_n for a positive integer n, are generated by 1 modulo n and have order n.^[29] Their presentation is \langle a \mid a^n = e \rangle, where e denotes the identity element.^[30] All cyclic groups are abelian, since powers of a single generator commute under the group operation.^[29] Free groups on r generators, where r is finite and at least 2, exemplify non-abelian finitely generated groups and serve as universal objects in group theory. The free group F_r on a generating set X with |X| = r consists of equivalence classes of words over the alphabet X \cup X^{-1}, where two words represent the same element if one can be obtained from the other by inserting or deleting subwords of the form xx^{-1} or x^{-1}x.^[31] Elements are represented by reduced words, which contain no such cancellable subwords, and the group operation is concatenation followed by reduction to obtain the unique reduced form.^[32] The rank r is well-defined, as any two free generating sets of F_r have the same cardinality.^[31] The defining universal property of F_r states that for any group G and any function f: X \to G, there exists a unique group homomorphism \phi: F_r \to G such that \phi(x) = f(x) for all x \in X.^[31] This property ensures F_r is the "freest" group generated by r elements, with no imposed relations among the generators beyond those required by the group axioms.^[32] A free basis of F_r is a generating set X such that every non-identity reduced word in X \cup X^{-1} is nontrivial in F_r.^[31] The automorphism group \mathrm{Aut}(F_r) consists of isomorphisms from F_r to itself and is generated by Nielsen transformations applied to an ordered basis: interchanging two basis elements, inverting a basis element, or replacing a basis element x_i with x_i x_j or x_i x_j^{-1} for i \neq j.^[33] These transformations enable the conversion of one free basis to another, preserving the group's structure and rank.^[33] As a concrete example, the free group F_2 on generators a and b has presentation \langle a, b \mid \rangle, imposing no relations.^[32] Its elements are reduced words formed by alternating positive and negative powers of a and b, such as a^3 b^{-1} a b^2, where no consecutive letters are inverses of each other.^[32] The Cayley graph of F_r with respect to a free basis X (taking S = X \cup X^{-1} as the symmetric generating set) is a $2r-regular tree, as the absence of relations implies no cycles in the graph.^[34] This tree structure arises because F_r acts freely and transitively on the graph's vertices.^[34] In comparison, the Cayley graph of the infinite cyclic group \mathbb{Z} with generator 1 is an infinite path (a bi-infinite line), while for the finite cyclic group \mathbb{Z}_n, it forms a cycle of length n.^[34]

Non-Abelian Examples

The symmetric group S_3 provides the smallest example of a finite non-abelian finitely generated group, consisting of all permutations of three elements and having order 6. It is generated by the adjacent transpositions (1\, 2) and (2\, 3), both of order 2, whose non-commutativity is evident since (1\, 2)(2\, 3) = (1\, 3\, 2) \neq (1\, 2\, 3) = (2\, 3)(1\, 2).^[35] This group admits the presentation \langle a, b \mid a^2 = b^3 = (ab)^2 = 1 \rangle, where a is a transposition of order 2 and b is a 3-cycle of order 3.^[36] Free products offer infinite non-abelian examples arising from combining simpler groups without additional relations. The free product \mathbb{Z} * \mathbb{Z}_2 is generated by an element x of infinite order from \mathbb{Z} and an element y of order 2 from \mathbb{Z}_2, yielding the presentation \langle x, y \mid y^2 = 1 \rangle; elements are reduced words alternating non-trivial powers of x and y. A related construction is the infinite dihedral group D_\infty, isomorphic to the free product \mathbb{Z}_2 * \mathbb{Z}_2 with presentation \langle t_1, t_2 \mid t_1^2 = t_2^2 = 1 \rangle, which contains an index-2 infinite cyclic subgroup generated by t_1 t_2 and models isometries of the real line.^[37] Surface groups exemplify non-abelian finitely generated groups tied to topology. The fundamental group of a closed orientable surface of genus g \geq 2 has the one-relator presentation

\langle a_1, b_1, \dots, a_g, b_g \mid \prod_{i=1}^g [a_i, b_i] = 1 \rangle,

where [a_i, b_i] = a_i b_i a_i^{-1} b_i^{-1} and there are $2g generators; these groups are non-abelian due to the non-trivial commutator relation and are hyperbolic, reflecting the negative Euler characteristic of the surface.^[38] The modular group \mathrm{SL}(2, \mathbb{Z}) is an arithmetic example of a non-abelian finitely generated group, generated by the matrices S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} of order 4 and T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} of infinite order. Its quotient \mathrm{PSL}(2, \mathbb{Z}) = \mathrm{SL}(2, \mathbb{Z}) / \{\pm I\} is the free product \mathbb{Z}_2 * \mathbb{Z}_3, amalgamated over the trivial subgroup, highlighting a tree-like structure in its Cayley graph.^[34] Non-abelian free groups F_r for r \geq 2 represent the freest non-abelian finitely generated groups, with presentation \langle x_1, \dots, x_r \mid \rangle imposing no relations beyond inverses. These groups surject onto every finite simple group through finite-index subgroups, enabling rich quotient structures that distinguish them from abelian cases.^[39]

Computational Aspects

Word Problem

In a finitely generated group G presented as \langle S \mid R \rangle, where S is a finite generating set and R is a finite set of relations, the word problem asks whether there exists an algorithm to determine, for any finite word w in the alphabet S \cup S^{-1}, if w represents the identity element of G.^[40] This problem, first formulated by Max Dehn in 1911, lies at the core of algorithmic group theory and connects computability with geometric and combinatorial structures.^[41] The word problem is solvable in free groups on a finite generating set S, where words reduce to a unique normal form via free reduction: repeatedly cancel adjacent inverse pairs until no such pairs remain, yielding the identity if and only if the empty word results.^[42] This process, a special case of Dehn's algorithm, terminates in linear time relative to the word length.^[43] Dehn originally developed his algorithm in 1912 to solve the word problem for fundamental groups of closed orientable surfaces of genus at least 2, using van Kampen diagrams and geodesic representatives in the hyperbolic plane to verify if a word equals the identity by checking for reducibility via relators.^[40] More broadly, a finitely generated group admits a Dehn algorithm—iteratively replacing subwords equal to relators or their inverses with the empty word until a geodesic (shortest) representative is obtained—if and only if it is word hyperbolic, as defined by Gromov in 1987; in such groups, the algorithm solves the word problem in linear time.^[44]^[45] Despite these positive cases, the word problem is undecidable for finitely presented groups in general. Independently, P. S. Novikov (1955) and W. W. Boone (1959) proved the existence of finitely presented groups where no algorithm can solve the word problem, by constructing presentations that embed computations of arbitrary Turing machines into the group's relations, reducing the halting problem to identity testing.^[40]^[46] This construction relies on Markov algorithms—normal algorithms over strings that simulate Turing machine steps—to encode non-halting computations as non-trivial words, ensuring undecidability via the known unsolvability of the halting problem.^[40] A. A. Markov's earlier 1947 work on undecidability for semigroup word problems provided the foundational technique of embedding recursive functions into algebraic relations. In contrast, the word problem is solvable for finitely generated abelian groups. Given a presentation \langle S \mid R \rangle of such a group, abelianize by adding commutator relations [s,t]=1 for all s,t \in S, yielding a presentation of \mathbb{Z}^n for some n; a word w then corresponds to an integer vector via exponent sums, and w equals the identity if and only if this vector is the zero vector, verifiable by linear algebra over \mathbb{Z}.^[47] This reduces to solving a system of linear equations, computable via the Smith normal form.^[48]

Growth and Isoperimetric Functions

In a finitely generated group G with respect to a finite symmetric generating set S, the growth function \beta_G(n) counts the number of elements g \in G such that the word length \|g\|_S \leq n, measuring how the group's "size" expands within balls of radius n in the Cayley graph.^[49] The asymptotic behavior of \beta_G(n) classifies groups into three main types: polynomial growth, where \beta_G(n) \sim n^d for some integer d \geq 0; exponential growth, where \beta_G(n) \sim e^{hn} for some h > 0; and intermediate growth, which is superpolynomial but subexponential.^[49] Groups exhibiting polynomial growth of degree d are precisely the virtually nilpotent ones, containing a finite-index nilpotent subgroup whose Hirsch length equals d, as established by Gromov's theorem. For example, the abelian group \mathbb{Z}^d has polynomial growth of exact degree d, reflecting the volume growth of balls in its Cayley graph, which is quasi-isometric to the Euclidean space \mathbb{R}^d.^[49] In contrast, free groups of rank at least 2 display exponential growth, with growth rate $2r - 1 where r is the rank with respect to a free basis, arising from the tree-like structure of their Cayley graphs.^[50] Intermediate growth occurs in groups like the Grigorchuk group, a finitely generated 2-group acting on the binary tree, whose growth function satisfies \exp(n^{c_1}) \prec \beta_G(n) \prec \exp(n^{c_2}) for constants $0 < c_1 < c_2 < 1, providing the first example resolving Milnor's question on the existence of such groups.^[51] Hyperbolic groups, introduced by Gromov, typically exhibit exponential growth and satisfy a linear isoperimetric inequality, meaning the Dehn function \delta(n) (defined below) is bounded by a linear function.^[44] The isoperimetric function, or Dehn function, of a finitely presented group G = \langle S \mid R \rangle quantifies the minimal "filling area" needed to span a loop of length at most n in the Cayley complex \tilde{K}(G, S, R), the universal cover of the presentation 2-complex, where \delta(n) is the supremum over all such loops of the minimal number of 2-cells required.^[52] This function links combinatorial word metrics to geometric filling properties, with \delta(n) independent of the finite presentation up to asymptotic equivalence.^[52] Gromov hyperbolic groups, characterized by \delta-thin triangles in their Cayley graphs (where side geodesics stay within \delta of each other), admit a linear isoperimetric inequality \delta(n) \preceq n, enabling efficient solutions to the word problem via Dehn's algorithm.^[44] The parameter \delta > 0 measures the "slimness" of triangles, with smaller \delta indicating stronger hyperbolicity, as in free groups where \delta = 0 due to their tree-like geometry.^[44]

Applications

Topology and Geometry

Finitely generated groups play a central role in topology and geometry as fundamental groups of manifolds and other spaces. The fundamental group \pi_1(M) of a connected manifold M encodes information about the loops in M up to homotopy and is finitely generated whenever M is compact, since compact manifolds have the homotopy type of finite CW-complexes whose fundamental groups are finitely generated by Seifert-van Kampen theorem applications.^[53] For example, the fundamental group of the 3-sphere S^3 is trivial, reflecting its simply connected nature, while that of the 2-torus T^2 is \mathbb{Z}^2, capturing its abelian structure from the product of circles. Aspherical spaces, where all higher homotopy groups vanish, provide classifying spaces K(G,1) for their fundamental group G, allowing geometric realization of group cohomology via topology. For finitely presented groups G, such models exist, and compact aspherical manifolds with fundamental group G are particularly significant in geometric group theory. Hyperbolic 3-manifolds exemplify this: they are aspherical, with finitely presented hyperbolic fundamental groups that admit discrete faithful representations into \mathrm{[PSL](/page/PSL)}(2,\mathbb{C}), enabling study of their geometry through Kleinian groups.^[54]^[55] Perelman's 2003 proof of Thurston's geometrization conjecture resolves the structure of 3-manifolds by showing that every compact orientable 3-manifold decomposes uniquely along incompressible tori into canonical pieces, each admitting one of eight Thurston geometries, with the fundamental group reflecting this JSJ (Jaco-Shalen-Johannson) decomposition. For finitely generated fundamental groups of such 3-manifolds, this implies a graph-of-groups decomposition where vertex and edge groups correspond to Seifert fibered or hyperbolic components, providing a topological invariant that decomposes the group along virtually abelian subgroups. Covering space theory establishes a Galois correspondence between connected covering spaces of a path-connected, locally path-connected space X and subgroups of \pi_1(X), with normal subgroups corresponding to regular (Galois) covers. In this context, finite generation is preserved: if \pi_1(X) is finitely generated, then the fundamental group of a finite-sheeted cover—corresponding to a finite-index subgroup—is also finitely generated, by Schreier's lemma bounding the number of generators.^[56] Knot groups illustrate these ideas concretely: the fundamental group of the complement of a knot K \subset S^3 is finitely presented, with abelianization \mathbb{Z} generated by the meridian class, and features a peripheral subgroup \mathbb{Z} \times \mathbb{Z} generated by the meridian and longitude, which remains unchanged under Dehn filling.^[57] This peripheral structure encodes knot invariants and relates to 3-manifold decompositions via covering spaces of the complement.

Number Theory and Algebraic Geometry

In number theory, finitely generated groups arise prominently in the study of arithmetic structures, such as Galois groups of field extensions and ideal class groups of rings of integers. These groups encode essential information about the arithmetic of number fields, including ramification and unit structures. In algebraic geometry, analogous constructions appear in the Picard groups of varieties and the étale fundamental groups of schemes, providing arithmetic invariants that parallel topological notions but adapted to the algebraic setting.^[58] The Kronecker-Weber theorem establishes that every finite abelian extension of the rational numbers \mathbb{Q} is contained in a cyclotomic extension, implying that the corresponding Galois groups are finite abelian and hence finitely generated.^[59] For a number field K, the Galois group of a finite Galois extension L/K is finite, thus finitely generated as an abstract group.^[58] In the context of abelian extensions, this finiteness ensures that the Galois groups capture the abelian structure without infinite generation requirements.^[59] The ideal class group of the ring of integers \mathcal{O}_K in a number field K is a finite abelian group, making it finitely generated.^[60] This finiteness, a cornerstone of algebraic number theory, measures the failure of unique factorization in \mathcal{O}_K.^[60] Dirichlet's unit theorem complements this by stating that the unit group \mathcal{O}_K^\times is finitely generated as an abelian group, with rank equal to r_1 + r_2 - 1, where r_1 is the number of real embeddings and $2r_2 is the number of complex embeddings of K.^[61] For quadratic fields, such as K = \mathbb{Q}(\sqrt{d}) with square-free integer d, the unit group \mathcal{Z}[\sqrt{d}]^\times has rank given precisely by this formula: rank 1 for real quadratic fields (d > 0) and rank 0 (torsion only) for imaginary quadratic fields (d < 0).^[61] In algebraic geometry, the Picard group \operatorname{Pic}(X) of an algebraic curve X over a field k classifies line bundles up to isomorphism. For an elliptic curve E over \mathbb{Q}, the Mordell-Weil theorem asserts that the group of rational points E(\mathbb{Q}) is finitely generated, and since \operatorname{Pic}^0(E) \cong E(\mathbb{Q}) as abelian varieties, the connected component of the Picard group is also finitely generated.^[62] The full Picard group \operatorname{Pic}(E) then decomposes as a finite extension of this finitely generated group by the Néron-Severi group, which is finitely generated for curves over number fields.^[62] The étale fundamental group \pi_1^{\text{ét}}(X, \overline{x}) of a connected scheme X of finite type over a separably closed field provides a profinite analogue of the topological fundamental group. For smooth projective varieties over algebraically closed fields, this group is topologically finitely generated, meaning it admits a dense finitely generated subgroup.^[63] Grothendieck's formulation draws an explicit analogy to topology, where finite étale covers correspond to quotients of the profinite completion of the fundamental group, ensuring finite generation in the profinite sense for schemes arising in arithmetic geometry.^[64]

Combinatorics and Cryptography

Finitely generated groups are central to combinatorial enumeration efforts, particularly through the study of group presentations, which specify a group via a finite set of generators and relations. A presentation \langle S \mid R \rangle defines a group as the quotient of the free group on S by the normal closure of R, allowing classification up to isomorphism via Tietze transformations that preserve the group structure. However, the isomorphism problem for finitely presented groups with at least two generators is undecidable, implying that there is no algorithm to enumerate all such groups up to isomorphism.^[65] The Burnside problem, posed in 1902, asks whether a finitely generated group in which every element has finite order (a periodic group) must be finite; counterexamples exist, such as the infinite finitely generated torsion groups constructed by Golod and Shafarevich in 1964 using Golod-Shafarevich groups, and later explicit examples like the Grigorchuk group and Gupta-Sidki p-groups.^[66]^[67] In graph theory, Cayley graphs of finitely generated groups provide a combinatorial framework for constructing expander graphs, which are sparse graphs with strong connectivity properties useful in derandomization. The Cayley graph \mathrm{Cay}(G, S) of a group G with symmetric generating set S has vertices as group elements and edges corresponding to multiplication by generators, enabling explicit constructions of expanders with spectral gaps bounded away from zero. For instance, Cayley graphs of non-abelian simple groups like \mathrm{PSL}(2, \mathbb{Z}/p\mathbb{Z}) yield families of expanders that derandomize probabilistic algorithms, such as those for approximate counting or pseudorandom generation, by providing deterministic substitutes for random walks.^[68] Finitely generated non-abelian groups feature prominently in cryptography, particularly in post-quantum protocols that resist attacks from quantum computers. The Anshel-Anshel-Goldfeld (AAG) key exchange protocol, introduced in 1999, uses braid groups—a finitely generated non-abelian family defined by Artin generators and relations—to enable secure key agreement via the difficulty of computing long braid words from short ones. In the protocol, parties select commuting subsets of braid elements as private keys and exchange public keys based on these subsets; the shared secret is derived from a commutator, whose computation relies on the conjugacy search problem in braid groups, which is presumed hard even when the word problem is solvable.^[69] This hardness stems partly from the intractability of the word problem in certain finitely presented groups, providing a security foundation for such schemes. The hidden subgroup problem (HSP) in finitely generated groups connects quantum computing to group theory, where one seeks to identify a hidden subgroup H \leq G given an oracle that reveals coset representatives. For abelian finitely generated groups, quantum algorithms using Fourier transforms solve HSP efficiently in polynomial time, underpinning applications like integer factorization via Shor's algorithm. In contrast, for non-abelian groups such as symmetric groups or matrix groups, no general efficient quantum algorithm exists, and the problem is believed to be hard, with partial solutions only for specific classes like wreath products or solvable groups; this difficulty motivates ongoing research into quantum-resistant cryptography based on non-abelian HSP instances.^[70]^[71] Polycyclic groups, a subclass of finitely generated solvable groups admitting a subnormal series with cyclic factors, admit efficient computational representations via polycyclic presentations (pc-presentations), which encode the group through a polycyclic generating sequence and power-commutator relations. These presentations enable polynomial-time algorithms for the word problem, subgroup computation, and factor group operations, making polycyclic groups practical for computational group theory applications like nilpotent quotient algorithms. For example, in GAP software, pc-presentations facilitate efficient handling of polycyclic-by-finite groups, supporting computations in algebraic number theory and representation theory.^[72]

Other Fields

In analysis, finitely generated abelian groups such as \mathbb{Z}^d serve as foundational examples for harmonic analysis, where the Fourier transform decomposes functions into characters of the dual group \mathbb{T}^d, the d-dimensional torus.^[73] This transform, defined for f \in \ell^1(\mathbb{Z}^d) as \hat{f}(\xi) = \sum_{n \in \mathbb{Z}^d} f(n) e^{-2\pi i \langle n, \xi \rangle} for \xi \in \mathbb{T}^d, enables the study of convolutions and inversion formulas, extending classical results to discrete settings and facilitating applications in signal processing on lattices.^[73] More broadly, harmonic analysis on such groups underpins the Plancherel theorem and spectral theory, with the Fourier transform acting unitarily on \ell^2(\mathbb{Z}^d).^[73] In probability theory, random walks on finitely generated groups provide models for diffusion processes, with recurrence and transience determined by group structure. Polya's theorem establishes that simple symmetric random walks on \mathbb{Z}^d are recurrent for d = 1, 2 (returning to the origin with probability 1 infinitely often) and transient for d \geq 3.^[74] Varopoulos' theorem generalizes this to arbitrary finitely generated groups, characterizing recurrence via the volume growth of Cayley balls: a group admits a recurrent random walk if and only if it has subexponential growth, linking probabilistic behavior to geometric properties.^[74] In physics, finitely generated groups model symmetries in crystal lattices through space groups, which are discrete subgroups of the Euclidean motion group acting co-compactly on \mathbb{R}^d. Bieberbach's theorems confirm that there are only finitely many such groups up to isomorphism in each dimension, with the translation subgroup isomorphic to \mathbb{Z}^d, ensuring finite generation.^[75] These groups capture periodic structures in solids, where the point group (finite rotational symmetries) acts on the lattice. Discretizations of Lie groups, such as finitely generated nilpotent subgroups approximating continuous symmetries, arise in numerical simulations of physical systems. A key example is the Heisenberg group, a three-dimensional nilpotent group generated by elements satisfying [x, y] = z with z central, modeling the canonical commutation relations in quantum mechanics for the harmonic oscillator.^[76] In chemistry, molecular symmetries typically involve finite point groups, but extensions to infinite discrete structures occur in periodic systems like coordination polymers and metal-organic frameworks (MOFs), where space group symmetries govern the lattice arrangements. These frameworks feature infinite chains or networks with translational symmetries generated by \mathbb{Z}^d-like lattices, enabling the design of porous materials with controlled porosity and conductivity. In biology, post-2000 developments have employed free groups and amalgamated products to model phylogenetic trees, capturing evolutionary relations through tree-like structures with branching via free amalgamations that represent speciation events. Free groups encode unresolved polytomies or reticulate evolution, while amalgamated free products model horizontal gene transfer by gluing subtrees along shared subgroups, providing algebraic tools for reconstructing ancestral relations from genomic data.^[77]

Finitely Presented Groups

A finitely presented group is a group that admits a finite presentation, meaning it can be expressed as the quotient of a free group on a finite generating set by the normal closure of a finite set of relators.^[78] This refines the notion of finite generation by imposing the additional constraint that the relations can also be encapsulated finitely, allowing for a compact algebraic description of the group's structure. Every finitely presented group is finitely generated, but the converse does not hold, as there exist finitely generated groups requiring infinitely many relators in any presentation.^[79] The deficiency of a finite presentation \langle X \mid R \rangle is defined as |X| - |R|, measuring the balance between generators and relators.^[80] The deficiency of the group itself is the supremum of these values over all finite presentations. For groups admitting a classifying space that is a finite CW-complex, the deficiency provides an upper bound for $1 minus the Euler characteristic:

\def(G) \leq 1 - \chi(G)

.^[81] This connection links combinatorial presentations to topological invariants, with equality often achieved in cases like free groups or surface groups. Examples of finitely presented groups abound in low-dimensional topology. The trivial group admits the presentation \langle a \mid a \rangle, with one generator and one relator imposing the identity. Fundamental groups of closed orientable surfaces of genus g \geq 1 are finitely presented; for genus g, they have a presentation with $2g generators a_1, b_1, \dots, a_g, b_g and a single relator \prod_{i=1}^g [a_i, b_i] = 1.^[82] In contrast, the restricted wreath product \mathbb{Z} \wr \mathbb{Z} is a canonical example of a two-generated group that is not finitely presented, as any presentation requires infinitely many relators to capture its structure.^[83] A profound property of finitely presented groups is the undecidability of their isomorphism problem: there is no algorithm to determine, given two finite presentations, whether they define isomorphic groups. This result, building on the Novikov-Boone theorem establishing undecidability of the word problem in some finitely presented groups, underscores the inherent computational complexity within this class.^[84]

Varieties of Groups

In group theory, a variety of groups is defined as a non-empty class of groups that is closed under the operations of taking subgroups, forming homomorphic images, and constructing arbitrary direct products.^[85] This closure property ensures that the class is stable under the basic constructions in group theory, making varieties a fundamental axiomatic framework for studying structural properties of groups.^[86] Birkhoff's theorem establishes that a class of groups forms a variety if and only if it can be defined by a set of group identities, or laws—equations of the form w(x_1, \dots, x_k) = 1, where w is a word in the free group on k generators, holding for all substitutions of group elements. For example, the variety of abelian groups is defined by the single identity xy = yx.^[85] More generally, varieties generated by finite groups are locally finite, meaning every finitely generated group in the variety is finite, as the laws imposed by the finite generators bound the order of elements.^[87] In the context of finitely generated groups, certain varieties impose restrictions that align with finite generation properties. The variety of nilpotent groups of class at most c is defined by identities ensuring that the (c+1)-th term of the lower central series is trivial, such as [[x_1, x_2], \dots, x_{c+1}] = 1 for appropriate commutators.^[85] Similarly, the variety of solvable groups of derived length at most k is axiomatized by identities making the (k+1)-th derived subgroup trivial.^[88] These varieties are finitely based when generated by finite sets of laws derivable from finite groups, and finitely generated groups within them exhibit controlled complexity in their series.^[87] A prominent example is the Burnside variety \mathcal{B}(n) of groups of exponent n, defined by the identity x^n = 1.^[89] The free Burnside group B(r, n) is then the relatively free group in this variety on r generators, consisting of all words in r letters modulo the laws of \mathcal{B}(n); for sufficiently large odd n \geq 4381 and r > 1, B(r, n) is infinite. Regarding discrimination within varieties, fully residually free groups—finitely generated groups where every non-trivial finite subset admits a homomorphism to a free group that is injective on the subset—arise in the study of the variety of all groups (with no non-trivial laws), serving as limit groups that embed densely in free group representations.^[90]

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Nov 7, 2019 · In the field of combinatorial group theory, Wilhelm Magnus studied representations of free groups by non-commutative power series (Lyndon and ...
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[PDF] arXiv:1705.02040v2 [math.GR] 18 Oct 2017
Oct 18, 2017 · ABSTRACT. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of ...
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Examples of non-finitely presented groups - Math Stack Exchange
May 9, 2014 · Examples of finitely generated, non-finitely presentable groups can be found via Rips' construction (E. Rips, Subgroups of small Cancellation Groups)definition - Is there difference between finitely presented groups and ...Non finitely presented subgroup - Mathematics Stack ExchangeMore results from math.stackexchange.com
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Finitely presented groups: The deficiency - ScienceDirect.com
mii, as i ranges over the natural ...
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Euler characteristics of 3-manifold groups and discrete subgroups of ...
Lower and upper bounds for χ(G) are given in terms of the rank and deficiency of G. ... The SQ-universality of some finitely presented groups. J. Austral. Math.
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One-relator surface groups
A one-relator group is just a free group modulo the normal closure of a single element. By analogy, a one-relator surface group is the fundamental group of a ...Missing: presentation | Show results with:presentation
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Rational subsets and submonoids of wreath products - ScienceDirect
The wreath product Z ≀ Z is one of the simplest examples of a finitely generated group that is not finitely presented, see [6], [7] for further results showing ...
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[PDF] arXiv:1005.2636v1 [math.LO] 14 May 2010
May 14, 2010 · Boone and Novikov [1, 8] independently showed that there exist groups with undecidable word problems, so this leads us to believe that ...
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variety of groups - PlanetMath
Mar 22, 2013 · Notes. By a theorem of Birkhoff [1] , a class of groups is a variety if and only if it is closed under taking subgroups , homomorphic images.
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Varieties of algebras - ScienceDirect.com
According to G. Birkhoff's theorem, varieties are exactly the non-empty classes of algebras of a given type which are closed under subalgebras, homomorphic ...
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Finitely generated equational classes - ScienceDirect.com
A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every subvariety of a finitely generated ...<|separator|>
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[PDF] Solvable group
More generally, all nilpotent groups are solvable. ... For any positive integer N, the solvable groups of derived length at most N form a subvariety of the ...<|separator|>
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[PDF] A Short Course on the Restricted Burnside Problem
Sep 15, 2025 · Definition 1.17. The free Burnside group B(r, n) is the free group in r generators for the variety of groups of exponent dividing n.
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[PDF] finitely presented residually free groups
For example, there are finitely generated subgroups of a direct product of two free groups for which the conjugacy problem and membership problem are unsolvable ...