In group theory, a branch of mathematics, given a groupG under a binary operation ∗, a subsetH 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 is a group operation on H. This is usually denoted, read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G. This is usually represented notationally by, read as "H is a proper subgroup of G". Some authors also exclude the trivial group from being proper. 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. The group G is sometimes denoted by the ordered pair, usually to emphasize the operation ∗ when G carries multiple algebraic or other structures.
Basic properties of subgroups
A subset H of the group G is a subgroup of Gif and only if it is nonempty and closed under products and inverses. In the case that H is finite, then H is a subgroup if and only if H is closed under products.
The above condition can be stated in terms of a homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism from H to G.
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.
The intersection of subgroups A and B is again a subgroup. The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane ; each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
If S is a subset of G, then there exists a minimum subgroup containing S, which can be found by taking the intersection of all of subgroups containing S; it is denoted by and is said to be the subgroup generated by S. An element ofG is in if and only if it is a finite product of elements of S and their inverses.
Every element a of a group G generates the cyclic subgroup. If is isomorphic to Z/nZ for some positive integern, then n is the smallest positive integer for which an = e, and n is called the order of a. If is isomorphic to Z, 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. If e is the identity of G, then the trivial group is the minimum subgroup of G, while the maximum subgroup is the group G itself.
File:Left cosets of Z 2 in Z 8.svg|thumb|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. Together they partition the entire group G into equal-size, non-overlapping sets. The index is 4.
Given a subgroup H and some a in G, we define the left cosetaH =. Because a is invertible, the map φ : H → aH given by φ = 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 ~ a2 if and only if a1−1a2 is in H. The number of left cosets of H is called the index of H in G and is denoted by . Lagrange's theorem states that for a finite groupG 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 must be a divisor of |G|. Right cosets are defined analogously: Ha =. They are also the equivalence classes for a suitable equivalence relation and their number is equal to . 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 is normal.
This group has two nontrivial subgroups: J= and H=, 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 group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Every group has as many small subgroups as neutral elements on the main diagonal: The and two-element groups Z2. These small subgroups are not counted in the following list.
12 elements
8 elements
6 elements
4 elements
3 elements
Other examples
The even integers are a subgroup of the additive group of integers: when you add two even numbers, you get an even number.
An ideal in a ring is a subgroup of the additive group of.
