Max-sensitive 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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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 $H$ ∩ $M$ 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.
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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).
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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.