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).
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.