# Join of all abelian normal subgroups

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

Suppose is a group. The **join of all abelian normal subgroups** of is the subgroup of defined as the join of *all* the abelian normal subgroups of .

If is a finite group (or more generally a group satisfying ascending chain condition on normal subgroups) then this is equivalent to the join of all subgroups that are maximal among abelian normal subgroups.