Minimal normal subgroup

This is a subgroup property obtained by applying the minimal operator to the trim subgroup property: {{{1}}}

History

The concept and terminology of minimal normal subgroups is part of the attempt to describe the structure of a group in terms of a successive chain of normal subgroups with simple quotients. The normal subgroup adjacent to the identity in this chain is the minimal normal subgroup.

Definition

Symbol-free definition

A nontrivial subgroup of a group is termed a minimal normal subgroup if it is normal and the only normal subgroup properly contained inside it is the trivial subgroup.

Definition with symbols

A nontrivial subgroup $H$ of a group $G$ is termed a minimal normal subgroup if it is normal and for any normal subgroup $K$ of $G$ such that $K$ ≤ $H$, either $K=H$ or $K$ is trivial.

In terms of the minimal operator

The property of being a minimal normal subgroup is an element of the subgroup property space, and is obtained by applying the minimal operator to the property of normality.

Metaproperties

Left realization

A group can be realized as a minimal normal subgroup of another group if and only if it is characteristically simple. This follows from the fact that characteristicity is the left transiter of normality.

Transitivity

A minimal normal subgroup of a minimal normal subgroup cannot be normal unless the minimal normal subgroup is a simple normal subgroup.

From the fact that every minimal normal subgroup is characteristically simple, and the fact that any minimal normal subgroup of a characteristically simple group is a simple direct factor, we conclude that any minimal normal subgroup of a minimal normal subgroup is actually simple.

This further tells us that though the property of being minimal normal is not transitive, the square of this property with respect to the composition operator equals its cube.

Intersection-triviality

This subgroup property is intersection-trivial, viz the intersection of two distinct subgroups satisfying the property must be trivial

The intersection of a minimal normal subgroup with any normal subgroup is either itself or trivial. In particular, the intersection of two distinct minimal normal subgroups is trivial. Thus, the property of being minimal normal is intersection-trivial.