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).
Definition
A finite p-group G is said to be regular if any of the following equivalent (Hall 1959, Ch. 12.4), (Huppert 1967, Kap. III §10) conditions are satisfied:
- 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 a · b = (ab) · c.
- For every a, b in G, there are elements ci in the derived subgroup of the subgroup generated by a and b, such that a · b = (ab) · c1 ⋯ ck.
- 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 a · b = (ab) · c1 ⋯ ck, where q = p.
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 p 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 p points is irregular and of order p.
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 Ωk(G) and regular groups are well-behaved in that Ωk(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 ℧k(G). In a regular group, the index 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+n().
- Philip Hall's criteria of regularity of a p-group G: G is regular, if one of the following hold:
- < p
- [G′:℧1(G′)| < p
- |Ω1(G)| < p
Generalizations
- Powerful p-group
- power closed p-group
References
- 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