Algebraic structure → Group theory Group theory 

In group theory, a branch of mathematics, 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}).^{[1]}^{[2]}
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.
Suppose that G is a group, and H is a subset of G.
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 a_{1} ~ a_{2} 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 G and H 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 G.^{[6]}^{[7]}
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.
Let G be the cyclic group Z_{8} whose elements are
and whose group operation is addition modulo 8. Its Cayley table is
+  0  4  2  6  1  5  3  7 

0  0  4  2  6  1  5  3  7 
4  4  0  6  2  5  1  7  3 
2  2  6  4  0  3  7  5  1 
6  6  2  0  4  7  3  1  5 
1  1  5  3  7  2  6  4  0 
5  5  1  7  3  6  2  0  4 
3  3  7  5  1  4  0  6  2 
7  7  3  1  5  0  4  2  6 
This group has two nontrivial subgroups: ■ J = {0, 4} and ■ H = {0, 4, 2, 6} , where J is also a subgroup of H. The Cayley table for H is the topleft quadrant of the Cayley table for G; The Cayley table for J is the topleft 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.
Let S_{4} be the symmetric group on 4 elements. Below are all the subgroups of S_{4}, listed according to the number of elements, in decreasing order.
The whole group S_{4} is a subgroup of S_{4}, of order 24. Its Cayley table is


Each element s of order 2 in S_{4} generates a subgroup {1,s} of order 2. There are 9 such elements: the transpositions (2cycles) and the three elements (12)(34), (13)(24), (14)(23).
The trivial subgroup is the unique subgroup of order 1 in S_{4}.