Upper join of characteristic subgroups

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Definition

A subgroup $H$ of a group $G$ is termed an upper join of characteristic subgroups if there exist subgroups $K_{i},i\in I$ of $G$ containing $H$ (where $I$ is an indexing set) such that:

• $H$ is a characteristic subgroup inside $K_{i}$.
• The join of the $K_{i}$s is $G$.

Note that $H$ need not be a characteristic subgroup of $G$ because characteristicity is not upper join-closed.

Formalisms

In terms of the upper join-closure operator

This property is obtained by applying the upper join-closure operator to the property: characteristic subgroup
View other properties obtained by applying the upper join-closure operator

Relation with other properties

Stronger properties

• Characteristic subgroup
• Commutator subgroup of direct factor
• Perfect normal subgroup

Weaker properties

• Normal subgroup