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
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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:

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

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

Relation with other properties

Stronger properties

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 $a, b \in G$ , $⟨ a, b ⟩$ is a regular $p$ -group, then $G$ itself is a regular $p$ -group.