# 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 $a^pb^p = (ab)^pc^p$.
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 $a^pb^p = (ab)^pc_1^pc_2^p \dots c_k^p$.
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 $a^qb^q = (ab)^qc_1^qc_2^q \dots c_k^q$ where $q = p^n$.

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

## 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.