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2 results

  • ON THE CENTRAL KERNEL OF A GROUP

  • Part of
  • ALESSIO RUSSO
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
    Published online by Cambridge University Press:
    13 October 2022, pp. 283-289
    Print publication:
    October 2023
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  • The central kernel $K(G)$ of a group G is the (characteristic) subgroup consisting of all elements $x\in G$ such that $x^{\gamma }=x$ for every central automorphism $\gamma $ of G. We prove that if G is a finite-by-nilpotent group whose central kernel has finite index, then the full automorphism group $Aut(G)$ of G is finite. Some applications of this result are given.

  • ON p-AUTOMORPHISMS THAT ARE INNER

  • Part of
  • M. SHABANI ATTAR
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 80 / Issue 2 / October 2009
    Published online by Cambridge University Press:
    08 June 2009, pp. 291-293
    Print publication:
    October 2009
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  • Let G be a group and let CAutΦ(G)(Z(Φ(G))) be the set of all automorphisms of G centralizing G/Φ(G) and Z(Φ(G)). For each prime p and finite p-group G, we prove that CAutΦ(G)(Z(Φ(G)))≤Inn(G) if and only if G is elementary abelian or Φ(G)=Z(G) and Z(G) is cyclic.