Extra-special p-groups as groups of automorphisms. Issue 2 (1st February 2023) - De Melo, E.; Gomes, M. E.; Lima, I.
Abstract: Let p be a prime and E an extra-special p -group of exponent p and order p 2 n + 1 . Suppose that E acts by automorphisms on a finite p ′ -group G . We prove that if γ n ( C G ( a ) ) is nilpotent for any a ∈ E #, then γ n ( G ) is nilpotent. If, for some integer d such that 2 d ≤ n, the d th derived group of C G ( a ) is nilpotent for any a ∈ E #, then the d th derived group G ( d ) is nilpotent. We also prove similar results for Lie algebras.