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
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.
- 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
|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.|