maximal subgroup

Let $G$ be a group.

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

Theorem.

A normal subgroup $N$ of a group $G$ is a maximal normal subgroupif and only if the quotient (http://planetmath.org/QuotientGroup) $G/N$ is simple (http://planetmath.org/Simple).

单词	MaximalSubgroup
释义	maximal subgroup Let $G$ be a group. A subgroup $H$ of $G$ is said to be a maximal subgroup of $G$ if $H\eq G$ and there is no subgroup $K$ of $G$ such 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 $G$ is said to be a maximal normal subgroup(or maximal proper normal subgroup) of $G$ if $N\eq G$ and there is no normal subgroup $K$ of $G$ such that $N<K<G$ .We have the following theorem: Theorem. A normal subgroup $N$ of a group $G$ is a maximal normal subgroupif and only if the quotient (http://planetmath.org/QuotientGroup) $G/N$ is simple (http://planetmath.org/Simple).
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