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 Phillip Hall (1934).
- For every a, b in G, there is a c in the derived subgroup H′ 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.
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.
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 (Huppert 1967, Kap III §10, Satz 10.8). 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
- [G′:℧1(G′)| < pp−1
- |Ω1(G)| < pp−1
- Hall, Marshall (1959), The theory of groups, Macmillan, MR 0103215
- Hall, Philip (1934), "A contribution to the theory of groups of prime-power order", Proceedings of the London Mathematical Society, 36: 29–95, doi:10.1112/plms/s2-36.1.29
- Huppert, B. (1967), Endliche Gruppen (in German), Berlin, New York: Springer-Verlag, pp. 90–93, ISBN 978-3-540-03825-2, MR 0224703, OCLC 527050