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: For every a, b in G, there is a c in the derived subgroup of the subgroup H of G generated by a and b, such that ap · bp = (ab)p · cp. For every a, b in G, there are elements ci in the derived subgroup of the subgroup generated by a and b, such that ap · bp = (ab)p · c1p ⋯ ckp. For every a, b in G and every positive integer n, there are elements ci in the derived subgroup of the subgroup generated by a and b such that aq · bq = (ab)q · c1q ⋯ ckq, where q = pn.
Examples Many familiar p-groups are regular: Every abelian p-group is regular. Every p-group of nilpotency class strictly less than p is regular. This follows from the Hall–Petresco identity. Every p-group of order at most pp is regular. Every finite group of exponent p is regular.
However, many familiar p-groups are not regular: Every nonabelian 2-group is irregular. The Sylow p-subgroup of the symmetric group on p2 points is irregular and of order pp+1.
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 pk is denoted Ωk(G) and regular groups are well-behaved in that Ωk(G) is precisely the set of elements of order dividing pk. The subgroup generated by all pk-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 Ωk(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]). Philip Hall's criteria of regularity of a p-group G: G is regular, if one of the following hold: [G:℧1(G)] < pp [:℧1()| < pp−1 |Ω1(G)| < pp−1
Generalizations
Powerful p-group power closed p-group
References