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On the wielandt subgroup in a p-group of maximal class

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Abstract

The Wielandt subgroup of a group G, denoted by w(G), is the intersection of the normalizers of all subnormal subgroups of G. In this paper, the authors show that for a p-group of maximal class G, either w i (G) = ζ i (G) for all integer i or w i (G) = ζ i+1(G) for every integer i, and w(G/K) = ζ(G/K) for every normal subgroup K in G with K ≠ 1. Meanwhile, a necessary and sufficient condition for a regular p-group of maximal class satisfying w(G) = ζ 2(G) is given. Finally, the authors prove that the power automorphism group PAut(G) is an elementary abelian p-group if G is a non-abelian p-group with elementary \(\zeta (G) \cap \mho _1 (G)\).

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Author information

Correspondence to Xiaohong Zhang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11071155) and the Key Disciplines of Shanghai Municipality (No. S30104).

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Zhang, X., Guo, X. On the wielandt subgroup in a p-group of maximal class. Chin. Ann. Math. Ser. B 33, 83–90 (2012). https://doi.org/10.1007/s11401-011-0690-z

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Keywords

  • p-Groups of maximal class
  • Wielandt subgroup
  • Wielandt series
  • Upper central series

2000 MR Subject Classification

  • 20D15
  • 20D25
  • 20D30