Submaximal 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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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed submaximal if there exists an ascending chain:

$H=H_{0}\leq H_{1}\leq H_{2}\leq \ldots \leq H_{n}=G$

where each $H_{i}$ is a maximal subgroup of $H_{i+1}$.

Formalisms

In terms of the subordination operator

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

Relation with other properties

Stronger properties

• Maximal subgroup
• Subgroup of finite index

Weaker properties

• Subpronormal subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity