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 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 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 · c1pckp.
  • 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 · c1qckq, where q = pn.

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 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:
    1. [G:℧1(G)] < pp
    2. [G:℧1(G)| < pp−1
    3. 1(G)| < pp−1

Generalizations

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
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