# Maximal subgroup

In mathematics, the term **maximal subgroup** is used to mean slightly different things in different areas of algebra.

In group theory, a **maximal subgroup** *H* of a group *G* is a proper subgroup, such that no proper subgroup *K* contains *H* strictly. In other words, *H* is a maximal element of the partially ordered set of proper subgroups of *G*. Maximal subgroups are of interest because of their direct connection with primitive permutation representations of *G*. They are also much studied for the purposes of finite group theory: see for example Frattini subgroup, the intersection of the maximal subgroups.

In semigroup theory, a **maximal subgroup** of a semigroup *S* is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) of *S* which is not properly contained in another subgroup of *S*. Notice that, here, there is no requirement that a maximal subgroup be proper, so if *S* is in fact a group then its unique maximal subgroup (as a semigroup) is *S* itself. Considering subgroups, and in particular maximal subgroups, of semigroups often allows one to apply group-theoretic techniques in semigroup theory. There is a one-to-one correspondence between idempotent elements of a semigroup and maximal subgroups of the semigroup: each idempotent element is the identity element of a unique maximal subgroup.

## Existence of maximal subgroup

Any proper subgroup of a finite group is contained in some maximal subgroup, since the proper subgroups form a finite partially ordered set under inclusion. There are, however, infinite Abelian groups that contain no maximal subgroups, for example the Prüfer group.[1]

## Maximal normal subgroup

Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup) of G if N<G and there is no normal subgroup K of G such that N<K<G. We have the following theorem:

**Theorem**: A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient G/N is simple.

## Hasse diagrams

These Hasse diagrams show the lattices of subgroups of S_{4}, Dih_{4} and Z_{2}^{3}.

The maximal subgroups are linked to the group itself (on top of the Hasse diagram) by an edge of the Hasse diagram.