In mathematics, given a group *G* under a binary operation *, we say that some subset *H* of *G* is 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* is a group operation on *H*.

A **proper subgroup** of a group *G* is a subgroup *H* which is a proper subset of *G* (i.e. *H* ≠ *G*). The **trivial subgroup** of any group is the subgroup {*e*} consisting of just the identity element.

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 (*G*,*), usually to emphasize the operation * when *G* carries multiple algebraic or other structures.

In the following, we follow the usual convention of dropping * and writing the product *a***b* as simply *ab*.

## Basic properties of subgroups

*H* is a subgroup of the group *G* if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever *a* and *b* are in *H*, then *ab* and *a*^{−1} are also in *H*. These two conditions can be combined into one equivalent condition: whenever *a* and *b* are in *H*, then *ab*^{−1} is also in *H*.) In the case that *H* is finite, it is enough that *H* is closed under products, since the closure under inverse follows automatically in that case.
- The identity of a subgroup is the identity of the group: if
*G* is a group with identity *e*_{G}, and *H* is a subgroup of *G* with identity *e*_{H}, then *e*_{H} = *e*_{G}.
- 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* = *e*_{H}, then *ab* = *ba* = *e*_{G}.
- If
*S* is a subset of *G*, then there exists a minimum subgroup containing *S*; it is denoted by <*S*> and is said to be 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.
- Every element
*a* of a group *G* generates the cyclic subgroup <*a*>. If <*a*> is isomorphic to **Z**/*n***Z** for some positive integer *n*, then *n* is the smallest positive integer for which *a*^{n} = *e*, and *n* is called the *order* of *a*. If <*a*> is isomorphic to **Z**, then *a* is said to have *infinite order*.
- The subgroups of any given group form a complete lattice under inclusion. (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.

## 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 *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} iff *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

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

where |*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: *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.

Last updated: 08-22-2005 12:08:17