Difference between revisions of "Submaximal subgroup"

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{{subgroup property}}
{{finitarily tautological subgroup property}}
{{finitarily tautological subgroup property}}

Latest revision as of 00:22, 8 May 2008

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]
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

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]


Definition with symbols

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

H = H_0 \le H_1 \le H_2 \le \ldots \le H_n = G

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


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

Weaker properties



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