Maximal subgroup

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

In other words, H is a maximal element of the partially ordered set of subgroups of G that are not equal to 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.

[citation needed] 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.

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: These Hasse diagrams show the lattices of subgroups of the symmetric group S4, the dihedral group D4, and C23, the third direct power of the cyclic group C2.

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