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" is denoted Ω("G") and regular groups are well-behaved in that Ω("G") is precisely the set of elements of order dividing "p". The subgroup generated by all "p"-th powers of elements in "G" is denoted ℧("G"). In  a regular group, the index [G:℧("G")] is equal to the order of Ω("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"),℧("N")] = ℧(["M","N"]).