Stem group

Definition
A stem group is defined as a group whose center is contained inside its derived subgroup. In symbols, a group $$G$$ is termed a stem group if $$Z(G) \le [G,G]$$ where $$Z(G)$$ denotes the center of $$G$$ and $$[G,G]$$ denotes the derived subgroup of $$G$$.

Stem groups are closely related to the concept of stem extensions. Specifically, any central extension where the resultant group is a stem group must be a stem extension. It is possible to have stem extensions where the resultant group is not a stem group. However, if the central extension has base normal subgroup the whole center, then the whole group is indeed a stem group.

Facts

 * Every group is isoclinic to a stem group
 * Stem group has the minimum order among all groups isoclinic to it
 * Formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

Original use

 * : Definition introduced on Page 135 (Page 6 of 12 relative to the paper).