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Paper   IPM / M / 11278
School of Mathematics
  Title:   Powerful p-groups have non-inner automorphisms of order p and some cohomology
  Author(s):  A. Abdollahi
  Status:   Published
  Journal: J. Algebra
  Vol.:  323
  Year:  2010
  Pages:   779-789
  Supported by:  IPM
In this paper we study the longstanding conjecture of whether there exists a non-inner automorphism of order p for a finite non-abelian p-group. we prove that if G is a finite non-abelian p-group such that G/Z(G) is powerful then G has a non-inner automorphism of order p leaving either Φ(G) or Ω1(Z(G)) elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for add p, by showing that the Tate cohomology Hn(G/N,Z(N)) ≠ 0 for all n\geqslant 0 , where G is a finite p-group, p is odd, G/Z(G) is p-central (i.e., elements of order p are central) and N\vartriangleleft G with G/N non-cyclic.

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