Maximal subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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History

The notion of maximal subgroup probably dates back to the very beginning of group theory.

Definition

Symbol-free definition

A maximal subgroup of a group is a proper subgroup such that there is no subgroup properly containing it and properly contained inside the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed maximal if $H$ is a proper subgroup of $G$ and there is no proper subgroup of $K$ that properly contains $H$ .

In terms of group actions

In terms of group actions, a subgroup of a group is maximal if the natural group action on its coset space is primitive.

Formalisms

In terms of the maximal operator

This property is obtained by applying the maximal operator to the property: subgroup
View other properties obtained by applying the maximal operator

Relation with other properties

Stronger properties

Weaker properties

NE-subgroup

Property theory

Closed or contraclosed

Given any expansive operator on the collection of subgroups of a group, a maximal subgroup is either closed with respect to that operator, or its closure is the whole group.

Every maximal subgroup is either normal or contranormal.
Every maximal subgroup is either characteristic or contracharacteristic.
Every maximal subgroup is either normal or self-normalizing.

Metaproperties

Transfer condition

In general, it may not be true that the intersection of a maximal subgroup with another subgroup is maximal inside that subgroup. If a subgroup has the property that its intersection with every maximal subgroup (not containing it) is maximal in it, the subgroup is termed max-sensitive.

Property operators

Transiters

The left and right transiters are both the identity element.

Subordination

The subordination property on the property of maximality defines the property of submaximality. For finite groups, every subgroup is submaximal. However, this may not be true in general for infinite groups. It is, however, true that every subgroup of finite index is submaximal.

The maximal subgroup operator

The maximal subgroup operator is a subgroup property operator that takes any subgroup property and gives otu the property of being a subgroup that is maximal in the group with respect to that property.