maximal subgroup

Let G be a group.

A subgroupMathworldPlanetmathPlanetmath H of G is said to be a maximal subgroup of G if HG and there is no subgroup K of G such that H<K<G. Note that a maximal subgroup of G is not maximal ( among all subgroups of G, but only among all proper subgroupsMathworldPlanetmath of G. For this reason, maximal subgroups are sometimes called maximal proper subgroups.

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


A normal subgroup N of a group G is a maximal normal subgroup if and only if the quotient ( G/N is simple (

Title maximal subgroup
Canonical name MaximalSubgroup
Date of creation 2013-03-22 12:23:46
Last modified on 2013-03-22 12:23:46
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Definition
Classification msc 20E28
Synonym maximal proper subgroup
Related topic MaximalElement
Defines maximal
Defines maximal normal subgroup
Defines maximal proper normal subgroup
Defines simplicity of quotient group