In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G , read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G ). This is often represented notationally by H < G , read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper (that is, H ≠ {e} ). If H is a subgroup of G, then G is sometimes called an overgroup of H. The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups.

Subgroup tests

Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition. a−1 is in H. These two conditions can be combined into one, that for every a and b in H, the element ab−1 is in H, but it is more natural and usually just as easy to test the two closure conditions separately. an−1 . If the group operation is instead denoted by addition, then closed under products should be replaced by closed under addition, which is the condition that for every a and b in H, the sum a + b is in H, and closed under inverses should be edited to say that for every a in H, the inverse −a is in H.

Basic properties of subgroups

eH = eG . ab = ba = eH , then ab = ba = eG . H → G sending each element a of H to itself is a homomorphism. A ⊆ B or B ⊆ A . A non-example: 2\Z \cup 3\Z is not a subgroup of \Z, because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in \R^2 is not a subgroup of \R^2. ⟨S⟩ and is called the subgroup generated by S. An element of G is in ⟨S⟩ if and only if it is a finite product of elements of S and their inverses, possibly repeated. ⟨a⟩ . If ⟨a⟩ is isomorphic to \Z/n\Z ([[Integers modulo n|the integers mod n ]]) for some positive integer n, then n is the smallest positive integer for which an = e , and n is called the order of a. If ⟨a⟩ is isomorphic to \Z, then a is said to have infinite order. {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself. Left cosets of Z 2 in Z 8.svg under addition. The subgroup H contains only 0 and 4, and is isomorphic to \Z/2\Z. There are four left cosets of H: H itself, 1 + H , 2 + H , and 3 + H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. The index [G : H] is 4.]]

Cosets and Lagrange's theorem

Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}. Because a is invertible, the map φ : H → aH given by φ(h) = ah is a bijection. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 if and only if a_1^{-1}a_2 is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H] . Lagrange's theorem states that for a finite group G and a subgroup H, where and denote the orders of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisor of. Right cosets are defined analogously: Ha = {ha : h in H}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H] . If aH = Ha for every a in G, then H is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.

Example: Subgroups of Z8

Let G be the cyclic group Z8 whose elements are and whose group operation is addition modulo 8. Its Cayley table is This group has two nontrivial subgroups: and , where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G; The Cayley table for J is the top-left quadrant of the Cayley table for H. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.

Example: Subgroups of S4

S4 is the symmetric group whose elements correspond to the permutations of 4 elements. Below are all its subgroups, ordered by cardinality. Each group (except those of cardinality 1 and 2) is represented by its Cayley table.

24 elements

Like each group, S4 is a subgroup of itself.

12 elements

The alternating group contains only the even permutations. It is one of the two nontrivial proper normal subgroups of S4 . (The other one is its Klein subgroup.) [[File:Alternating group 4; Cayley table; numbers.svg|thumb|left|323px|Alternating group A4 Subgroups: ]]

8 elements

6 elements

4 elements

3 elements

2 elements

Each permutation p of order 2 generates a subgroup {1, p} . These are the permutations that have only 2-cycles:

1 element

The trivial subgroup is the unique subgroup of order 1.

Other examples