Stem group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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) \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

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).