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Homemaximal subgroup

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# maximal subgroup

Let $G$ be a group.

A subgroup $H$ of $G$
is said to be a *maximal subgroup* of $G$
if $H\neq G$ 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 subgroups of $G$.
For this reason, maximal subgroups are sometimes called *maximal proper subgroups*.

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

Defines:

maximal, maximal normal subgroup, maximal proper normal subgroup, simplicity of quotient group

Related:

MaximalElement

Synonym:

maximal proper subgroup

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

20E28*no label found*

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