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

Suppose $p$ is a prime number. A p-group $G$ (i.e., a group where the order of every element is a power of $p$) is termed a regular $p$-group if it satisfies the following equivalent conditions:

1. For every $a,b\in G$, there exists $c\in [\langle a,b\rangle ,\langle a,b\rangle ]$ such that $a^{p}b^{p}=(ab)^{p}c^{p}$.
2. For every $a,b\in G$, there exist $c_{1},c_{2},\dots ,c_{k}\in [\langle a,b\rangle ,\langle a,b\rangle ]$ such that $a^{p}b^{p}=(ab)^{p}c_{1}^{p}c_{2}^{p}\dots c_{k}^{p}$.
3. For every $a,b\in G$ and every natural number $n$, there exist $c_{1},c_{2},\dots ,c_{k}\in [\langle a,b\rangle ,\langle a,b\rangle ]$ such that $a^{q}b^{q}=(ab)^{q}c_{1}^{q}c_{2}^{q}\dots c_{k}^{q}$ where $q=p^{n}$.

The term regular p-group is typically used only for finite p-groups.

Relation with other properties

Stronger properties

• Abelian p-group
• Odd-order class two p-group
• p-group of nilpotency class less than p

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	regular p-group property is subgroup-closed	If $p$ is prime, $G$ is a regular $p$-group, and $H$ is a subgroup of $G$, then $H$ is a regular $p$-group.
quotient-closed group property	Yes	regular p-group property is quotient-closed	If $p$ is prime, $G$ is a regular $p$-group, $H$ is a normal subgroup of $G$, and $G/H$ is the corresponding quotient group, then $G/H$ is also a regular $p$-group.
finite direct product-closed group property	No	regular p-group property is finite direct product-closed	It is possible to have a prime number $p$ and regular $p$-groups $G_{1},G_{2}$ such that the external direct product $G_{1}\times G_{2}$ is not a regular $p$-group.
2-local group property	Yes	regular p-group property is 2-local	Suppose $p$ is a prime number and $G$ is a group such that for all $a,b\in G$, $\langle a,b\rangle$ is a regular $p$-group, then $G$ itself is a regular $p$-group.