Upper join of characteristic subgroups

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Upper join of characteristic subgroups, all facts related to Upper join of characteristic subgroups) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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