Stem group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Contents
Definition
A stem group is defined as a group whose center is contained inside its derived subgroup. In symbols, a group is termed a stem group if where denotes the center of and denotes the derived subgroup of .
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 link}^{More info}: Definition introduced on Page 135 (Page 6 of 12 relative to the paper).