Ingroup theory, a branch of mathematics, given a groupG under a binary operation ∗, a subsetHofG is called a subgroupofGifH 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.[1]
Aproper subgroup of a group G is a subgroup H which is a proper subsetofG (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}).[2][3]
IfH is a subgroup of G, then G is sometimes called an overgroupofH.
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. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition.
Then H is a subgroup of Gif and only ifH is nonempty and closed under products and inverses. Closed under products means that for every a and binH, the product ab is in H. Closed under inverses means that for every ainH, the inverse a−1 is in H. These two conditions can be combined into one, that for every a and binH, the element ab−1 is in H, but it is more natural and usually just as easy to test the two closure conditions separately.[4]
When Hisfinite, the test can be simplified: H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element aofH generates a finite cyclic subgroup of H, say of order n, and then the inverse of aisan−1.[4]
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 binH, the sum a + b is in H, and closed under inverses should be edited to say that for every ainH, the inverse −a is in H.
The identity of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG.
The inverse of an element in a subgroup is the inverse of the element in the group: if H is a subgroup of a group G, and a and b are elements of H such that ab = ba = eH, then ab = ba = eG.
IfH is a subgroup of G, then the inclusion map H → G sending each element aofH to itself is a homomorphism.
The intersection of subgroups A and BofG is again a subgroup of G.[5] For example, the intersection of the x-axis and y-axis in under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
The union of subgroups A and B is a subgroup if and only if A ⊆ BorB ⊆ A. A non-example: is not a subgroup of 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 is not a subgroup of
IfS is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by ⟨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.[6]
Every element a of a group G generates a cyclic subgroup ⟨a⟩. If ⟨a⟩isisomorphicto (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 orderofa. If ⟨a⟩ is isomorphic to then a is said to have infinite order.
The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual set-theoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union itself.) If e is the identity of G, then the trivial group {e} is the minimum subgroup of G, while the maximum subgroup is the group G itself.
G is the group the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to 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.
Given a subgroup H and some ainG, we define the left cosetaH = {ah : hinH}. 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 relationa1 ~ a2if and only if is in H. The number of left cosets of H is called the indexofHinG and is denoted by [G : H].
where |G| and |H| denote the ordersofG and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a divisorof|G|.[7][8]
Right cosets are defined analogously: Ha = {ha : hinH}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [G : H].
IfaH = Ha for every ainG, 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.
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 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 Giscyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.[9]
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.
under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.
is not a subgroup of 
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 
(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 orderofa. If ⟨a⟩ is isomorphic to 
the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to 
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.
![{\displaystyle [G:H]={|G| \over |H|}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e64b76d0605e6c7891273e65da28b5e431d4ea4)
of the integer ring 
the sum of two even integers is even, and the negative of an even integer is even.