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

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