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

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).