Regular p-group

In mathematical finite group theory, the concept of regular "p"-group captures some of the more important properties of abelian "p"-groups, but is general enough to include most "small" "p"-groups. Regular "p"-groups were introduced by .
Definition.
A finite "p"-group "G" is said to be regular if any of the following equivalent , conditions are satisfied:
Examples.
Many familiar "p"-groups are regular:
However, many familiar "p"-groups are not regular:
Properties.
A "p"-group is regular if and only if every subgroup generated by two elements is regular.
Every subgroup and quotient group of a regular group is regular, but the direct product of regular groups need not be regular.
A 2-group is regular if and only if it is abelian. A 3-group with two generators is regular if and only if its derived subgroup is cyclic. Every "p"-group of odd order with cyclic derived subgroup is regular.
The subgroup of a "p"-group "G" generated by the elements of order dividing "p""k" is denoted Ω"k"("G") and regular groups are well-behaved in that Ω"k"("G") is precisely the set of elements of order dividing "p""k". The subgroup generated by all "p""k"-th powers of elements in "G" is denoted ℧"k"("G"). In a regular group, the index [G:℧"k"("G")] is equal to the order of Ω"k"("G"). In fact, commutators and powers interact in particularly simple ways . For example, given normal subgroups "M" and "N" of a regular "p"-group "G" and nonnegative integers "m" and "n", one has [℧"m"("M"),℧"n"("N")] = ℧"m"+"n"(["M","N"]).