Direct factor of fully invariant subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: direct factor and fully characteristic subgroup
View other such compositions|View all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a direct factor of fully invariant subgroup if it satisfies the following equivalent conditions:

• It can be expressed as a direct factor of a fully invariant subgroup.
• It is a direct factor in its fully invariant closure.

In terms of the composition operator

The subgroup property of being a direct factor of fully invariant is obtained by applying the composition operator to the subgroup properties of being a direct factor and of being fully invariant.

Relation with other properties

Stronger properties

• Fully invariant subgroup
• Direct factor

Weaker properties

• Direct factor of characteristic subgroup
• Central factor of characteristic subgroup
• Normal subgroup of characteristic subgroup
• 2-subnormal subgroup
• Subnormal subgroup