In mathematics, given a group

*G*under an operation *, we say that some subset

*H*of

*G*is a

**subgroup**if

*H*is a group under * also. (The same definition applies more generally when

*G*is an arbitrary semigroup, but this article will only deal with subgroups of groups.)

It is easily shown that *H* is a subgroup of the group *G* if and only if it is nonempty and closed to products and inverses.
Furthermore, *H*'s identity element is equal to *G*'s identity element, and the inverse of an element of *H* is the same as the inverse of that element in *G*.

The subgroups of any given group form a complete lattice under inclusion. There is a minimal subgroup, the trivial group {*e*} (*e* being *G*'s identity element), and a maximal subgroup, the group *G* itself.

If *S* is a subset of *G*, then there exists a minimal subgroup containing *S*; it is denoted by <*S*> and is said to be * generated* by *S*. The elements of <*S*> are all finite products of elements of *S* and their inverses. Groups generated by a single element are called *cyclic* and are isomorphic to either (**Z**, +), where **Z** denotes the integers, or to (**Z**_{n}, +), where **Z**_{n} denotes the integers modulo *n* for some positive integer *n* (see modular arithmetic).

**Order of an element of a group**: Given an element *x* of *G*, the order of the cyclic subgroup *order* of *x*; it is the smallest positive integer *n* such that *x*^{n} = *e*.

Given a subgroup *H* and some *g* in G, we define the *left coset* *g***H* = {*g*h* : *h* in *H*}. Because *g* is invertible, the set *g*H* has just as many elements as *H*. 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 *g*_{1} ~ *g*_{2} iff *g*_{1}^{-1} * *g*_{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

- [
*G*:*H*] |*H*| = |*G*|

*G*| and |

*H*| denote the cardinalities of

*G*and

*H*, respectively. In particular, if

*G*is finite, then the cardinality of every subgroup of

*G*(and the order of every element of

*G*) must be a divisor of |

*G*|.

*Right cosets* are defined analogously: *H*g* = {*h*g* : *h* in *H*}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [*G* : *H*].
If *g*H* = *H*g* for every *g* in *G*, then *H* is said to be a normal subgroup. In that case we can define a multiplication on cosets by

- (
*g1*H*)*(*g2*H*) := (*g1*g2*)**H*

*G/H*. There is a natural homomorphism

*f*:

*G*

`->`

*G/H*given by

*f*(

*g*)=

*g*H*. The image

*f*(

*H*) consists only of the identity element of

*G/H*, the coset

*e*H*.

In general, a group homomorphism *f*: *G* `->` *K* sends subgroups of *G* to subgroups of *K*. Also, the preimage of any subgroup of *K* is a subgroup of *G*. We call the preimage of the trivial group {*e*} in *K* the *kernel* of the homomorphism and denote it by ker(*f*). As it turns out, the kernel is always normal and the image *f*(*G*) of *G* is always isomorphic to *G*/ker(*f*).

The normal subgroups of any group *G* form a lattice under inclusion. The minimal and maximal elements are {*e*} and *G*, the greatest lower bound of two subgroup is their intersection and their least upper bound is a product group.