# 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 $H$ $M$ 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.
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.