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Inmathematics, the termmaximal subgroup is used to mean slightly different things in different areas ofalgebra.
Ingroup theory, amaximal subgroupH of agroupG is aproper subgroup, such that no proper subgroupK containsH strictly. In other words,H is amaximal element of thepartially ordered set of subgroups ofG that are not equal toG. Maximal subgroups are of interest because of their direct connection withprimitive permutation representations ofG. They are also much studied for the purposes offinite group theory: see for exampleFrattini subgroup, the intersection of the maximal subgroups.
Insemigroup theory, amaximal subgroup of a semigroupS is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) ofS which is not properly contained in another subgroup ofS. Notice that, here, there is no requirement that a maximal subgroup be proper, so ifS is in fact a group then its unique maximal subgroup (as a semigroup) isS itself. Considering subgroups, and in particular maximal subgroups, of semigroups often allows one to apply group-theoretic techniques in semigroup theory.[citation needed] There is a one-to-one correspondence betweenidempotent elements of a semigroup and maximal subgroups of the semigroup: each idempotent element is theidentity element of a unique maximal subgroup.
Any proper subgroup of a finite group is contained in some maximal subgroup, since the proper subgroups form a finitepartially ordered set under inclusion. There are, however, infiniteabelian groups that contain no maximal subgroups, for example thePrüfer group.
Similarly, anormal subgroupN ofG is said to be a maximal normal subgroup (or maximal proper normal subgroup) ofG ifN <G and there is no normal subgroupK ofG such thatN <K <G. We have the following theorem:
TheseHasse diagrams show thelattices of subgroups of thesymmetric groupS4, thedihedral groupD4, andC23, the thirddirect power of thecyclic group C2.
The maximal subgroups are linked to the group itself (on top of the Hasse diagram) by an edge of the Hasse diagram.
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