Stem group

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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 $Z(G)\leq [G,G]$ 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

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
centerless group	the center is the trivial subgroup			|FULL LIST, MORE INFO
perfect group	the derived subgroup is the whole group			|FULL LIST, MORE INFO
non-abelian nilpotent UL-equivalent group	non-abelian nilpotent group whose upper central series and lower central series coincide.

References

Journal references

Original use

• The classification of prime-power groups by Philip Hall, Volume 69, (Year 1937): ^{Official linkMore info}: Definition introduced on Page 135 (Page 6 of 12 relative to the paper).