Absolutely regular p-group

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 p-group $G$ is termed an absolutely regular p-group if $|G/\mho ^{1}(G)|\leq p^{p-1}$, where $\mho ^{1}(G)$ denotes the first agemo subgroup of $G$, i.e., the subgroup of $G$ generated by $g^{p},g\in G$.

Relation with other properties

Stronger properties

• Group of prime order
• Cyclic group of prime power order

Weaker properties

• Regular p-group: For proof of the implication, refer absolutely regular implies regular and for proof of its strictness (i.e. the reverse implication being false) refer regular not implies absolutely regular.