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Regular p group

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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).

Contents

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 · 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 (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. |Ω1(G)| < pp−1

    • Generalizations

  • Powerful p-group
  • power closed p-group

    • References

    Regular p-group Wikipedia
    (Text) CC BY-SA