Subgroup
In group theory, a subgroup of a group G is a nonempty subset H of G that forms a group under the same binary operation as G.[1] To qualify as a subgroup, H must satisfy closure under the operation (if a, b \in H, then ab \in H), contain the identity element of G, and include the inverse of every element in H (if a \in H, then a^{-1} \in H); associativity is inherited from G.[1] The notation H \leq G indicates that H is a subgroup of G.[2] Subgroups provide essential insights into the structure of groups by allowing the study of smaller groups within larger ones, facilitating classifications and decompositions.[2] Trivial subgroups include the singleton containing only the identity element \{e\} and the entire group G itself, while proper subgroups exclude these cases.[1] The intersection of any collection of subgroups of G is itself a subgroup, and the smallest subgroup containing a given subset X \subseteq G, denoted \langle X \rangle, is generated by finite products of elements from X and their inverses.[2] Key types of subgroups include cyclic subgroups, generated by a single element and isomorphic to either the infinite cyclic group or a finite cyclic group of order dividing the generator's order; normal subgroups, which are invariant under conjugation by elements of G (i.e., gHg^{-1} = H for all g \in G) and enable the formation of quotient groups; and Sylow p-subgroups, which are maximal subgroups of order a power of a prime p and are central to the Sylow theorems for classifying finite groups.[2] Subgroups underpin fundamental results such as Lagrange's theorem, which states that the order of a subgroup divides the order of the group, and play crucial roles in homomorphisms, group extensions, representation theory, and applications to geometry and symmetry.[2]Definition and Fundamentals
Definition of a Subgroup
In group theory, a subgroup of a group G is a non-empty subset H \subseteq G that is closed under the group operation and under taking inverses. Formally, for all a, b \in H, the product a \cdot b \in H, and for all a \in H, the inverse a^{-1} \in H.[3] The requirement that H is non-empty ensures the presence of the identity element e of G. To see this, let h \in H; since H is closed under inverses, h^{-1} \in H, and thus h \cdot h^{-1} = e \in H. With the identity in H and closure under the operation, H inherits the group structure from G.[3] Subgroups are classified as proper if H \neq G. The trivial subgroup \{e\} contains only the identity element, and G itself is always a subgroup, called the improper subgroup.[3]Notation and Conventions
In group theory, the standard notation for indicating that H is a subgroup of a group G is H \leq G.[4] This symbol emphasizes the algebraic structure beyond mere set inclusion, as H \subseteq G alone does not guarantee the subgroup properties.[5] For the subgroup generated by a subset S of G, the common notation is \langle S \rangle or hSi, representing the smallest subgroup containing S.[6] When dealing with group orders, the cardinality |H| denotes the number of elements in a finite subgroup H, while infinite groups lack this finite measure and are not assigned an order in the same way.[7] Groups are typically denoted as (G, *), where * is the binary operation, though additive notation may be used for abelian groups like (\mathbb{Z}, +).[8] Not every subset of a group qualifies as a subgroup, as the subset must satisfy closure, identity inclusion, and inverses under the group operation. For instance, the even integers form a subgroup of (\mathbb{Z}, +) but do not constitute a group under multiplication, since the multiplicative identity 1 is odd and not included.[6] Special cases include the trivial subgroup \{e\}, consisting solely of the identity element e, and the improper subgroup G itself, which is always a subgroup but excluded from considerations of proper subgroups.[8]Subgroup Criteria
One-Step Subgroup Test
The one-step subgroup test provides a streamlined criterion for verifying whether a nonempty subset H of a group G forms a subgroup. Specifically, H is a subgroup of G if and only if a^{-1}b \in H for all a, b \in H.[9] This condition combines aspects of closure and inverses into a single verification step, leveraging the group's existing structure. To see why this implies the standard subgroup properties, first note that the operation in [H](/page/H+) inherits associativity from [G](/page/G). Setting a = b yields a^{-1}a = e \in [H](/page/H+), confirming the identity is in [H](/page/H+). For inverses, fix a \in [H](/page/H+) and set b = e, so a^{-1}e = a^{-1} \in [H](/page/H+). For closure under the group operation, take a, b \in [H](/page/H+); since b^{-1} \in [H](/page/H+), it follows that a b = a (b^{-1})^{-1} \in [H](/page/H+) by the test applied to a and b^{-1}.[9] Thus, all subgroup axioms hold. This test is particularly efficient for subsets where computing or verifying a^{-1}b is straightforward, such as in finite groups where elements can be enumerated or in settings with natural inversion operations like matrix groups. It applies equally to infinite groups but may be less practical there if the subset lacks a simple description under the inverse-difference operation.[9] A representative example is the set of rotations in the dihedral group D_n (for n \geq 3), which consists of elements \{ e, r, r^2, \dots, r^{n-1} \}, where r is rotation by $2\pi/n. To verify using the one-step test, take a = r^i and b = r^j in this set; then a^{-1}b = r^{-i} r^j = r^{j-i}, which is also a power of r between 0 and n-1, hence in the set. Thus, the rotations form a subgroup isomorphic to the cyclic group of order n.[10]Two-Step Subgroup Test
The two-step subgroup test is a criterion for verifying that a nonempty subset H of a group G forms a subgroup by separately confirming closure under the group operation and closure under inverses.[11] Specifically, H \leq G if for all a, b \in H, ab \in H, and for all a \in H, a^{-1} \in H.[11] This approach contrasts with more unified tests by emphasizing distinct verifications, which aids in building intuitive understanding of subgroup properties.[11] The presence of the identity element e of G in H follows directly from these conditions. Since H is nonempty, select any h \in H; then h^{-1} \in H by inverse closure, and h \cdot h^{-1} = e \in H by operational closure.[9] Associativity inherits from G, completing the subgroup structure.[9] This test proves effective for manual verification in small or finite sets, where checking each property individually clarifies failures and successes without requiring advanced machinery.[11] A counterexample illustrating the need for inverse closure is the set of positive real numbers in the additive group (\mathbb{R}, +). This set is nonempty and closed under addition, as the sum of positives remains positive, but it lacks inverses, since the additive inverse of any positive real is negative and outside the set.[12]Core Properties
Closure and Identity in Subgroups
A subgroup H of a group G is defined such that it is closed under the group operation, meaning that for all h_1, h_2 \in H, the product h_1 \cdot h_2 also belongs to H.[2] This closure property ensures that H remains invariant under repeated applications of the operation from G, forming a self-contained algebraic structure within G.[13] For instance, if h \in H, then the positive powers h^n for n \in \mathbb{N} are obtained by successive multiplications and thus lie in H, highlighting how closure propagates the operation iteratively.[2] The identity element e_G of the parent group G serves as the identity e_H for the subgroup H, as it satisfies e_G \cdot h = h \cdot e_G = h for all h \in H, inheriting this role directly from G's structure.[13] This shared identity underscores the embedding of H within G, requiring no separate verification since any subgroup must include e_G to maintain group axioms.[2] Consequently, powers of elements extend to all integers n \in \mathbb{Z}, with h^n \in H for h \in H, as negative exponents incorporate the inverse operation alongside closure.[13] Associativity in H is automatically inherited from G, as the operation on H is merely the restriction of G's associative binary operation, eliminating the need for separate checks.[2] This inheritance preserves the algebraic consistency essential to group theory. Regarding finite generation, closure facilitates the construction of subgroups generated by a subset X \subseteq G, denoted \langle X \rangle, which comprises all finite products derived from elements of X and their inverses under the operation, forming the smallest subgroup containing X.[13] For a single element x \in G, the cyclic subgroup \langle x \rangle = \{ x^n \mid n \in \mathbb{Z} \} exemplifies this, built solely through closure and the identity.[2]Inverses and Order Preservation
A fundamental property of subgroups is their closure under the inverse operation: if [H](/page/H+) is a subgroup of a group G and h \in [H](/page/H+), then the inverse h^{-1} must also belong to [H](/page/H+).[6] This ensures that [H](/page/H+) forms a group under the same operation as G, allowing every element in [H](/page/H+) to pair with its inverse within [H](/page/H+) to yield the identity element. Furthermore, the inverse operation in subgroups inherits the general group property that the inverse of an inverse recovers the original element: if h \in H, then (h^{-1})^{-1} = h.[14] To see this, note that h^{-1} \cdot h = e and h \cdot h^{-1} = e by the group axioms; thus, h acts as a left and right inverse for h^{-1}. Since inverses are unique in any group, it follows that (h^{-1})^{-1} = h, and both h and h^{-1} remain in H.[14] This bidirectional closure reinforces the self-contained nature of subgroups. Subgroups also preserve the orders of their elements relative to the ambient group. Specifically, if a \in H \subseteq G and the order of a in G is n (the smallest positive integer such that a^n = e), then the order of a in H is also n, as the cyclic subgroup \langle a \rangle = \{ e, a, a^2, \dots, a^{n-1} \} is contained in H due to closure under the operation.[15] More precisely, since the powers of a up to n generate the relations defining the order, and all such powers lie in H, no smaller exponent yields the identity in H than in G. This equality holds because the minimal exponent is determined by the element's action alone, independent of the larger group structure. In finite groups, this order preservation has structural implications: the orders of elements in H must divide the order of G, teasing the deeper result that |H| divides |G|, though the full proof involves cosets.[15] For instance, consider the rotation subgroup of the dihedral group D_5, which models symmetries of a regular pentagon and consists of rotations R_0, R_1, R_2, R_3, R_4 (adding 0 through 4 modulo 5). The inverse of R_1 is R_4, since R_1 \circ R_4 = R_5 = R_0 (the identity), and both remain in the rotation subgroup; similarly, R_2^{-1} = R_3.[16] This illustrates inverse closure while preserving the cyclic order of 5 for generators like R_1.Cosets and Structural Theorems
Left and Right Cosets
In group theory, given a group G and a subgroup H \leq G, the left coset of H generated by an element a \in G is the set aH = \{ ah \mid h \in H \}.[17] Similarly, the right coset is Ha = \{ ha \mid h \in H \}.[18] These sets represent translates of H under left or right multiplication by a, respectively, and each has the same cardinality as H.[17] Two left cosets aH and bH are equal if and only if a^{-1}b \in H.[17] The analogous condition holds for right cosets: Ha = Hb if and only if ab^{-1} \in H.[18] Moreover, distinct cosets are disjoint; that is, if aH \neq bH, then aH \cap bH = \emptyset.[17] The collection of all left cosets of H in G forms a partition of G, meaning G is the disjoint union of these cosets, each of size |H|.[17] The same holds for right cosets.[18] If G is finite, the number of distinct left (or right) cosets, denoted [G : H], equals |G| / |H|.[17] A subgroup H is normal in G if and only if every left coset of H equals the corresponding right coset, i.e., aH = Ha for all a \in G.[17] This equivalence holds automatically in abelian groups but requires additional structure otherwise.[18]Lagrange's Theorem and Consequences
Lagrange's theorem is a fundamental result in group theory that relates the orders of a finite group and its subgroups. It states that if G is a finite group and H is a subgroup of G, then the order of H, denoted |H|, divides the order of G, denoted |G|.[19] Additionally, the index of H in G, written [G : H], which is the number of distinct left (or right) cosets of H in G, equals |G| / |H|.[20] The proof relies on the partition of G into disjoint left cosets of H. Each coset gH = \{ gh \mid h \in H \} for g \in G has exactly |H| elements, as the map h \mapsto gh is a bijection by left cancellation in groups.[19] Distinct cosets are disjoint, and their union is G, so |G| equals the number of cosets times |H|, implying |H| divides |G| and [G : H] = |G| / |H|.[20] A key corollary is that the order of any element g \in G divides |G|, since the cyclic subgroup \langle g \rangle generated by g has order equal to the order of g, which must divide |G| by Lagrange's theorem.[20] Consequently, g^{|G|} = e for all g \in G, where e is the identity.[20] Another important consequence is that if H is a proper subgroup of another subgroup K \leq G, then |K| \geq 2 |H|, because K is a disjoint union of at least two cosets of H in K, so no subgroup of G can have order strictly between |H| and $2 |H|.[21] Lagrange's theorem has significant applications in number theory. For instance, Fermat's Little Theorem—that if p is prime and a is an integer not divisible by p, then a^{p-1} \equiv 1 \pmod{p}—follows as a special case. The multiplicative group (\mathbb{Z}/p\mathbb{Z})^\times of units modulo p has order p-1, so by the corollary on element orders, the order of _p divides p-1, yielding _p^{p-1} = {{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}_p.[22] While powerful for finite groups, Lagrange's theorem does not extend to infinite groups in the same way, as subgroup orders need not "divide" the group's cardinality in a constraining manner. For example, the additive group \mathbb{Z} of integers has subgroups m\mathbb{Z} for each positive integer m, all of which are infinite and have finite index m, but the theorem's divisibility condition on finite orders does not apply or impose similar restrictions.[21]Illustrative Examples
Subgroups of the Cyclic Group ℤ₈
The cyclic group \mathbb{Z}_8 is the additive group of integers modulo 8, consisting of the elements \{0, 1, 2, 3, 4, 5, 6, 7\} with addition modulo 8.[23] As a finite cyclic group of order 8, its subgroups are precisely the cyclic subgroups generated by elements whose orders divide 8, and there is exactly one subgroup for each divisor of 8.[23] The trivial subgroups are the identity subgroup \{0\} of order 1 and the full group \mathbb{Z}_8 itself of order 8.[23] The proper nontrivial subgroups are \langle 4 \rangle = \{0, 4\} of order 2 and \langle 2 \rangle = \{0, 2, 4, 6\} of order 4; note that \langle 1 \rangle = \mathbb{Z}_8, \langle 3 \rangle = \mathbb{Z}_8, \langle 5 \rangle = \mathbb{Z}_8, \langle 7 \rangle = \mathbb{Z}_8, \langle 6 \rangle = \langle 2 \rangle, and \langle 0 \rangle = \{0\}.[23] There are no subgroups of order 3 or 5, as the possible orders of subgroups must divide 8 by Lagrange's theorem.[23] All subgroups of \mathbb{Z}_8 are cyclic, as subgroups of cyclic groups are always cyclic, and they are generated by multiples of the generator corresponding to the divisors of 8 (specifically, generated by k where k = 8/d for each divisor d of 8).[23] The subgroup lattice of \mathbb{Z}_8 forms a chain reflecting the divisor lattice of 8: \{0\} \subset \langle 4 \rangle \subset \langle 2 \rangle \subset \mathbb{Z}_8, where each inclusion is proper and the structure arises from the unique subgroups ordered by inclusion corresponding to the divisors 1, 2, 4, and 8.[24]| Subgroup | Generator | Order | Elements |
|---|---|---|---|
| Trivial | \langle [0](/page/0) \rangle or \langle 8 \rangle | 1 | \{[0](/page/0)\} |
| Order 2 | \langle 4 \rangle | 2 | \{0, 4\} |
| Order 4 | \langle 2 \rangle | 4 | \{0, 2, 4, 6\} |
| Full group | \langle [1](/page/1) \rangle | 8 | \{0, 1, 2, 3, 4, 5, 6, 7\} |