Regular p-group

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.

Stronger properties

 * Weaker than::Abelian p-group
 * Weaker than::Odd-order class two p-group
 * Weaker than::p-group of nilpotency class less than p