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Commutator subgroup

In group theory, the commutator subgroup of a group G, commonly denoted G' or [G, G], is the smallest subgroup generated by all commutators [a, b] = a^{-1}b^{-1}ab for elements a, b \in G.^[1] This subgroup serves as a measure of the "non-abelianness" of G, as it is the unique smallest normal subgroup such that the quotient group G / [G, G] is abelian.^[2] For abelian groups, the commutator subgroup is trivial, consisting only of the identity element.^[3] The commutator subgroup plays a central role in the study of group structure and homological algebra. It is always normal in G, and the quotient G / G' is known as the abelianization of G, which captures the maximal abelian quotient and is isomorphic to the first homology group H_1(G, \mathbb{Z}).^[1] A group G is termed perfect if G = G', meaning it equals its own commutator subgroup, as seen in simple non-abelian groups like the alternating group A_n for n \geq 5.^[3] Notable examples include the symmetric group S_n (for n \geq 3), where the commutator subgroup is the alternating group A_n, and the general linear group \mathrm{GL}(n, \mathbb{R}) over the reals, where [\mathrm{GL}(n, \mathbb{R}), \mathrm{GL}(n, \mathbb{R})] = \mathrm{SL}(n, \mathbb{R}), the special linear group.^[1]^[3] Further properties highlight its significance: not every element of G' need be a single commutator, but the subgroup is generated by them, and it facilitates the construction of the derived series G \triangleright G' \triangleright G'' \triangleright \cdots, which terminates at the trivial group for solvable groups.^[2] This structure is foundational in classifying finite groups and understanding extensions, with applications extending to Lie groups and algebraic topology.^[3]

Commutator Elements

Definition of Commutator

In group theory, the commutator of two elements a, b in a group G (with multiplicative operation and inverses) is defined as [a, b] = a^{-1} b^{-1} a b.^[4] This expression quantifies the extent to which a and b fail to commute under the group operation.^[5] The commutator equals the identity element e if and only if a and b commute, meaning ab = ba.^[4] Some references instead define the commutator as [a, b] = a b a^{-1} b^{-1}, which is the inverse of the standard form and yields equivalent algebraic properties up to inversion.^[6] The notation [a, b] = a^{-1} b^{-1} a b is standard in many abstract algebra texts.^[6] For instance, in an abelian group where all elements commute, every commutator is the identity.^[4] In the non-abelian symmetric group S_3 on three letters, consider the transpositions a = (1\, 2) and b = (1\, 3); then [a, b] = (1\, 2\, 3), a 3-cycle. Similarly, [(1\, 3), (2\, 3)] = (1\, 3\, 2).

Properties of Individual Commutators

The commutator of any group element with itself is the identity element. Specifically, for any a in a group G, [a, a] = a^{-1} a^{-1} a a = e, where e is the identity, following directly from the group axioms of inverses and associativity.^[7] Similarly, the commutator with the identity is trivial: [a, e] = a^{-1} e^{-1} a e = a^{-1} a = e.^[7] The inverse of a commutator is the commutator of the reversed arguments: ([a, b])^{-1} = [b, a]. This follows from direct computation using the definition [a, b] = a^{-1} b^{-1} a b, yielding (a^{-1} b^{-1} a b)^{-1} = b^{-1} a^{-1} b a = [b, a], again relying on the uniqueness of inverses in groups.^[8] Commutators capture deviations from ideal conjugation behavior in groups. In particular, the conjugate of b by a satisfies a b a^{-1} = b [a, b]^a, where [a, b]^a = a [a, b] a^{-1} denotes conjugation by a; this relation arises by rearranging a b = b a [a, b] and conjugating, highlighting how non-commutativity introduces a conjugated commutator factor.^[9] Equivalently, in the reversed conjugation, b^{-1} a b = a [a, b], expressing the conjugate of a by b as a adjusted by the commutator itself.^[7] Key identities govern the behavior of commutators with products and inverses, derivable from the group axioms. For instance, [a, bc] = [a, c] [a, b]^c, where ^c denotes conjugation by c. To derive this, expand [a, bc] = a^{-1} (bc)^{-1} a (bc) = a^{-1} c^{-1} b^{-1} a b c; inserting c c^{-1} = e after a c yields a^{-1} c^{-1} a c \cdot c^{-1} a^{-1} b^{-1} a b c = [a, c] [a, b]^c.^[7] Likewise, [ab, c] = [a, c]^b [b, c], obtained by expanding [ab, c] = (ab)^{-1} c^{-1} (ab) c = b^{-1} a^{-1} c^{-1} a b c and regrouping terms using b b^{-1} = e to separate the contributions: b^{-1} (a^{-1} c^{-1} a c) b \cdot b^{-1} c^{-1} b c = [a, c]^b [b, c].^[10] For inverses, [a, b^{-1}] = [b, a]^b, where g^b = b g b^{-1} denotes conjugation by b; this follows from direct computation: [b, a]^b = b (b^{-1} a^{-1} b a) b^{-1} = a^{-1} b a b^{-1} = [a, b^{-1}].^[7] In free groups, commutators exhibit bilinearity up to conjugation, meaning expressions like [a_1^{k_1} \cdots a_m^{k_m}, b_1^{l_1} \cdots b_n^{l_n}] expand into sums (products) of basic commutators modulated by conjugations, a property foundational to the associated Lie ring structure but distinct from general groups.^[11] Products of commutators follow limited expansion rules without necessarily simplifying to single commutators; for example, using the above identities, [a, b] [c, d] can be manipulated via conjugations, but in general requires higher-order terms unless additional relations hold in the group.^[9]

Definition of the Subgroup

Generation and Construction

The commutator subgroup G' of a group G, also denoted [G, G], is defined as the subgroup generated by all commutators of elements in G, that is, G' = \langle [g, h] \mid g, h \in G \rangle, where [g, h] = g^{-1} h^{-1} g h. This is the smallest subgroup of G containing every commutator, consisting of all finite products of commutators and their inverses. In groups where the set of commutators is infinite, such as certain infinite groups, G' may require an infinite generating set of commutators, though each element of G' is a finite product from this set.^[12]^[13]^[14] The closure of G' under the group operation and inversion follows from its construction as a subgroup generated by the commutators. Specifically, the inverse of any commutator [g, h]^{-1} = [h, g], which is itself a commutator, ensuring that inverses remain within the generating set. Products of commutators are included by the definition of the generated subgroup, as any finite product of elements from the generating set (commutators and their inverses) lies in G'. These properties rely on the basic identities of commutators, such as the alternation under inversion.^[14]^[2]^[15] In the context of universal algebra, the commutator subgroup G' is the verbal subgroup of G corresponding to the binary word w(x, y) = [x, y], meaning it is generated by all substitutions of group elements into this word. This verbal characterization highlights G' as the fully invariant subgroup defined by the commutator law.^[16] Regarding finiteness, if G is a finite group, then G' is finitely generated, as the finite set of all commutators in G serves as a finite generating set for G'. In contrast, for infinite groups, G' need not be finitely generated, even if G is finitely generated.^[17]^[18]

Normality and Abelian Quotient

The commutator subgroup G' of a group G, defined as the subgroup generated by all commutators [a, b] = a^{-1} b^{-1} a b for a, b \in G, is normal in G. To see this, consider any g \in G and commutator [a, b] \in G'. Conjugation yields g [a, b] g^{-1} = g a^{-1} b^{-1} a b g^{-1} = (g a g^{-1})^{-1} (g b g^{-1})^{-1} (g a g^{-1}) (g b g^{-1}) = [g a g^{-1}, g b g^{-1}], which is itself a commutator and thus lies in G'. Since G' is generated by commutators, conjugation by g maps G' into itself, establishing normality.^[2] The quotient group G / G' is abelian. For any a G', b G' \in G / G', their product satisfies (a G') (b G') = a b G' and (b G') (a G') = b a G', so a b G' = b a G' if and only if a b (b a)^{-1} = a b a^{-1} b^{-1} = [a, b]^{-1} \in G', which holds by definition of the cosets. Thus, multiplication in the quotient commutes, making G / G' abelian.^[2] The quotient G / G' satisfies the universal property of the abelianization of G, denoted G^{\mathrm{ab}}. Specifically, for any homomorphism \phi: G \to A where A is abelian, there exists a unique homomorphism \overline{\phi}: G / G' \to A such that \phi = \overline{\phi} \circ \pi, with \pi: G \to G / G' the canonical projection. To construct \overline{\phi}, define \overline{\phi}(g G') = \phi(g); this is well-defined because if g G' = h G', then g h^{-1} \in G', and since A is abelian, G' \subseteq \ker \phi, so \phi(g) = \phi(h). Homomorphism properties follow from those of \phi, and uniqueness holds as any such map must agree on cosets. This establishes G^{\mathrm{ab}} \cong G / G'.^[14] The commutator subgroup G' is the intersection of the kernels of all homomorphisms from G to abelian groups. Any such homomorphism \phi: G \to A (with A abelian) has G' \subseteq \ker \phi, since \phi([g, h]) = \phi(g) \phi(h) \phi(g)^{-1} \phi(h)^{-1} = 1 in A, and thus \phi vanishes on the subgroup generated by commutators. Conversely, if g \notin G', the coset g G' is nontrivial in the abelian group G / G', so there exists a homomorphism \chi: G / G' \to S^1 (or another abelian group) with \chi(g G') \neq 1, and the composition G \to G / G' \to S^1 is a homomorphism to an abelian group with g \notin \ker(\chi \circ \pi). By Zorn's lemma, the intersection of all such kernels is precisely G'.^[19]

Key Properties and Relations

Abelianization

The abelianization of a group G, denoted G^{\mathrm{ab}} or \mathrm{Ab}(G), is the quotient group G/G', where G' is the commutator subgroup of G. This construction defines a covariant functor \mathrm{Ab}: \mathbf{Grp} \to \mathbf{Ab} from the category of groups to the category of abelian groups, which sends each group homomorphism f: G \to H to the induced homomorphism \overline{f}: G/G' \to H/H' on the quotients. The functor \mathrm{Ab} is left adjoint to the inclusion functor i: \mathbf{Ab} \hookrightarrow \mathbf{Grp}, meaning that for any group G and abelian group A, there is a natural isomorphism \mathrm{Hom}_{\mathbf{Grp}}(G, i(A)) \cong \mathrm{Hom}_{\mathbf{Ab}}(\mathrm{Ab}(G), A).^[20] This adjunction captures the universal property of the abelianization: any homomorphism \phi: G \to A to an abelian group A factors uniquely through the projection \pi: G \to G^{\mathrm{ab}} as \overline{\phi} \circ \pi = \phi, where \overline{\phi}: G^{\mathrm{ab}} \to A is the unique induced map. The functor preserves direct products, satisfying (G \times H)^{\mathrm{ab}} \cong G^{\mathrm{ab}} \times H^{\mathrm{ab}} for groups G and H.^[21] To compute the abelianization explicitly, consider a presentation G = \langle X \mid R \rangle, where X is a generating set and R is a set of relators. The abelianization G^{\mathrm{ab}} is then isomorphic to the abelian group \mathbb{Z}^{(X)} / N, where \mathbb{Z}^{(X)} is the free abelian group on X and N is the subgroup generated by the abelianized relators r^{\mathrm{ab}} for each r \in R (obtained by replacing the group operation with addition and commutators with zero). For instance, if G is the free group on a finite set X with |X| = n, then G^{\mathrm{ab}} \cong \mathbb{Z}^n, so the free abelian rank equals the number of generators. This method reduces the computation to linear algebra over \mathbb{Z}, such as finding the Smith normal form of the relation matrix.^[22] The abelianization also connects to algebraic topology and homology theory. Specifically, the first integral homology group H_1(G, \mathbb{Z}) of a discrete group G (with trivial \mathbb{Z}G-action on \mathbb{Z}) is isomorphic to G^{\mathrm{ab}}. This isomorphism arises from the bar resolution or other projective resolutions of \mathbb{Z} over \mathbb{Z}G, where the zeroth homology is \mathbb{Z} and the first homology captures the cycles modulo boundaries, equivalent to the coinvariants G/[G,G]. For a discrete group G, this corresponds topologically to the fact that H_1(BG, \mathbb{Z}) \cong \pi_1(BG)^{\mathrm{ab}} = G^{\mathrm{ab}}, where BG is the classifying space of G.^[23]

Derived Series

The derived series of a group G is defined recursively as the descending chain of subgroups G^{(0)} = G, G^{(1)} = G', where G' is the commutator subgroup of G, and G^{(n+1)} = (G^{(n)})' for each integer n \geq 0.^[4] Each term G^{(n)} is normal in G, as the derived subgroup of a group is characteristic in it, and characteristic subgroups of normal subgroups are normal in the ambient group; this property propagates through the iteration.^[4] The series forms a decreasing sequence G^{(0)} \supseteq G^{(1)} \supseteq G^{(2)} \supseteq \cdots, with each factor G^{(n)} / G^{(n+1)} abelian by construction, as G^{(n+1)} is generated by commutators within G^{(n)}.^[4] A group G is solvable if and only if its derived series terminates at the trivial subgroup, meaning there exists a finite integer k such that G^{(k)} = \{e\}; the minimal such k is called the derived length of G.^[4] To see this, suppose G^{(k)} = \{e\}; then \{e\} = G^{(k)} \trianglelefteq G^{(k-1)} \trianglelefteq \cdots \trianglelefteq G^{(1)} \trianglelefteq G^{(0)} = G forms a subnormal series (with each term normal in the previous) whose successive quotients G^{(i)} / G^{(i+1)} are abelian, establishing solvability by the standard definition via abelian factors.^[4] Conversely, if G admits a subnormal series with abelian factors, the derived series can be shown to reach the trivial subgroup in finitely many steps, as each abelian quotient collapses commutators in the subsequent terms.^[4] For infinite groups, the derived series may require a transfinite extension, indexed by ordinals beyond the finite naturals, to fully capture iterated commutators until stabilization or triviality; however, solvability is primarily characterized in the finite case where termination occurs after finitely many steps.^[24] In nilpotent groups, which are always solvable, the derived series terminates at the trivial subgroup, but it does so more rapidly than the lower central series, highlighting the distinction between solvability (measured by the derived series) and nilpotency (measured by central extensions via the lower central series).^[25]

Connections to Group Classes

The commutator subgroup distinguishes perfect groups, defined as those groups G for which G' = G.^[26] This equality implies that G has no nontrivial abelian quotients, as the abelianization G/G' is trivial.^[26] A representative example is the special linear group \mathrm{SL}_n(\mathbb{C}) for n \geq 2, which is perfect.^[26] Solvable groups are characterized by the property that the derived series, starting with G^{(0)} = G and G^{(i+1)} = (G^{(i)})' , terminates at the trivial subgroup after finitely many steps, so G^{(k)} = \{e\} for some k \geq 1. For such groups, the commutator subgroup G' is a proper subgroup of G. In contrast, nonsolvable groups like the alternating group A_5 have derived series that do not reach the trivial subgroup. Metabelian groups form a subclass where the second derived subgroup G'' = \{e\}, meaning the commutator subgroup G' is abelian.^[27] Supersolvable groups refine the solvable class by admitting a normal series with cyclic factors; in these groups, the commutator subgroup is nilpotent. Polycyclic groups, possessing subnormal series with cyclic factors, relate closely to supersolvable groups, though not all finite polycyclic groups are supersolvable (for example, the alternating group A_4).^[28] In boundedly generated groups, the commutator length—the minimal number of commutators needed to express any element of G'—admits bounds, particularly in contexts like perfect groups where the commutator width is studied.^[29]

Examples and Applications

Concrete Group Examples

The commutator subgroup of the symmetric group S_n for n \geq 3 is the alternating group A_n.^[1] This follows from the fact that A_n is generated by 3-cycles, each of which can be expressed as a commutator in S_n; for distinct a, b, c, the 3-cycle (a\, b\, c) equals the commutator [(a\, b), (a\, c)].^[1] Since the 3-cycles generate A_n and A_n has index 2 in S_n, it is the unique normal subgroup of that index, confirming that the commutator subgroup coincides with A_n.^[1] For the alternating group A_n, the commutator subgroup is A_n itself when n \geq 5, making A_n a perfect group.^[1] In this case, every even permutation can be generated from commutators within A_n, leveraging the simplicity of A_n for n \geq 5. For n=4, however, A_4' = V_4, the Klein four-group consisting of the identity and the three double transpositions.^[1] In the general linear group \mathrm{GL}_n(\mathbb{R}) for n \geq 2, the commutator subgroup is the special linear group \mathrm{SL}_n(\mathbb{R}). This is established by the determinant homomorphism \det: \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^*, which has kernel \mathrm{SL}_n(\mathbb{R}) and abelian image \mathbb{R}^*, so the abelianization of \mathrm{GL}_n(\mathbb{R}) is \mathbb{R}^* and the commutator subgroup is precisely the kernel. The Heisenberg group over \mathbb{Z} consists of $3 \times 3 upper triangular matrices with 1s on the diagonal and integer entries above, under matrix multiplication. Its commutator subgroup is the center, which is the subgroup of matrices with nonzero entry only in the (1,3)-position, isomorphic to \mathbb{Z}.^[30] The abelianization is then isomorphic to \mathbb{Z}^2, reflecting the nilpotency class 2 structure.^[31] For any abelian group G, the commutator subgroup is trivial, i.e., G' = \{e\}, since all commutators are the identity.^[1]

Maps Involving Outer Automorphisms

The commutator subgroup G' of a group G is characteristic, meaning that every automorphism of G preserves G' setwise.^[32] Consequently, every automorphism \phi \in \Aut(G) induces an automorphism on the quotient G/G', the abelianization of G, via the map \phi_* : gG' \mapsto \phi(g)G'.^[33] This yields a natural homomorphism \Aut(G) \to \Aut(G/G'), which descends to a homomorphism \Out(G) \to \Out(G/G') upon quotienting by the inner automorphisms.^[33] Since G/G' is abelian, its inner automorphism group is trivial, so \Out(G/G') \cong \Aut(G/G').^[34] The kernel of the map \Out(G) \to \Aut(G/G') consists of the classes of outer automorphisms that act trivially on the abelianization; these are the images of the IA-automorphisms, which fix G/G' pointwise.^[35] This kernel captures how outer automorphisms of G interact with the derived structure without altering the abelian invariants. A prominent example occurs for the free group F_n of rank n \geq 2, where the abelianization is \mathbb{Z}^n and the induced map is \Out(F_n) \to \GL_n(\mathbb{Z}).^[35] This map is surjective, realized by Nielsen transformations, which generate \Aut(F_n) and include operations like swapping generators, inverting a generator, and multiplying one generator by another.^[36] The kernel, known as the Torelli subgroup, encodes the non-abelian aspects of automorphisms of free groups.^[35] In various groups, such as the symmetric group S_n for n \geq 3, the abelianization is \mathbb{Z}/2\mathbb{Z} and \Aut(\mathbb{Z}/2\mathbb{Z}) is trivial, so the map is surjective (trivially).^[34] More generally, this map detects compatibility in group extensions: for an extension $1 \to A \to E \to G \to 1 with abelian kernel A, the induced action of G on A factors through the abelianization, and outer automorphisms of G must preserve this to lift.^[34] Surjectivity holds in cases like free groups, allowing full realization of linear actions on abelian quotients.^[35] This structure relates to group cohomology, where for a fixed action of G on an abelian module A, the second cohomology group H^2(G, A) classifies extensions up to equivalence, with the outer action of G on A determining the compatibility of automorphisms.^[37]

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