maximal subgroup (original) (raw)

Let G be a group.

A subgroup H of Gis said to be a maximal subgroup of Gif H≠G and there is no subgroup K of Gsuch that H<K<G. Note that a maximal subgroup of G is not maximal (http://planetmath.org/MaximalElement) among all subgroups of G, but only among all proper subgroups of G. For this reason, maximal subgroups are sometimes called maximal proper subgroups.

Similarly, a normal subgroup N of Gis said to be a maximal normal subgroup(or maximal proper normal subgroup) of Gif N≠G and there is no normal subgroup K of Gsuch 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 (http://planetmath.org/QuotientGroup) G/Nis simple (http://planetmath.org/Simple).