Max-sensitive subgroup

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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]

Definition

Symbol-free definition

A subgroup of a group is termed max-sensitive if its intersection with any maximal subgroup of the whole group is either equal to it or a maximal subgroup of it.

Definition with symbols

A subgroup H of a group G is termed max-sensitive in G if, for any maximal subgroup M of G, the group HM is either the whole of H or a maximal subgroup of H.

In terms of the subgroup intersection restriction formalism

The property of being max-sensitive is the balanced subgroup property in the subgroup intersection restriction formalism corresponding to the subgroup property of being a maximal 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

The subgroup property of being max-sensitive is transitive, on account of being a balanced subgroup property with respect to a restriction formalism.

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The subgroup property of being max-sensitive is identity-true, that is, every group is max-sensitive as a subgroup of itself.

It is also trivially true, that is, the trivial subgroup is always a max-sensitive subgroup.