Join of all abelian normal subgroups

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

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

If $G$ 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.

Facts

• Join of all abelian normal subgroups is isoclinism-invariant