Finite p-group in which the number of nth roots is a power of p for all n

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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 finite p-group in which the number of nth roots is a power of p for all n if a finite p-group in which, for any fixed n \in \mathbb{N}, the number of solutions to:

x^n = e

is a power of the prime p.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian group of prime power order
Group of prime power order 1-isomorphic to an abelian group
Finite regular p-group
Finite Lazard Lie group
Group of prime power order order statistics-equivalent to an abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite group in which all cumulative order statistics values divide the order of the group