On outer automorphisms of Cernikov p-groups
On outer automorphisms of Cernikov p-groups
This paper concerns the existence of outer p-automorphisms in Chernikov p-groups. The author proves that, if G is a non-nilpotent Chernikov p- group such that the Fitting subgroup F properly contains the finite residual \(G_ 0\) and the intersection of \(G_ 0\) with the centre Z(G) of G is divisible, then G has an outer p-automorphism. Moreover, if \(F=G_ 0\) and Z(G) is divisible, then either G has an outer p- automorphism or \(H^ 1(G/G_ 0,G_ 0)=0\) and the natural image of \(G/G_ 0\) in Aut \(G_ 0\) is a Sylow p-subgroup of Aut \(G_ 0\). The last section of the paper is devoted to the construction of some examples showing that there exist groups of this type without outer p- automorphisms. In his proofs, the author uses previous results about the existence of outer p-automorphisms in nilpotent p-groups ([\textit{W. Gaschütz}, J. Algebra 4, 1-2 (1966; Zbl 0142.260)], [\textit{P. Schmid}, Math. Z. 147, 271-277 (1976; Zbl 0307.20016)], [\textit{F. Menegazzo} and \textit{S. E. Stonehewer}, J. Lond. Math. Soc., II. Ser. 31, 272-276 (1985; Zbl 0526.20023)], [\textit{R. Marconi}, Rend. Semin. Mat. Univ. Padova 74, 123-127 (1985; Zbl 0588.20027)]).
- University of Florence Italy
Automorphisms of infinite groups, Sylow p-subgroup, outer automorphism, Fitting subgroup, Periodic groups; locally finite groups, Chernikov p-groups, Subgroup theorems; subgroup growth, finite residual, Automorphism groups of groups, outer p-automorphisms, centre
Automorphisms of infinite groups, Sylow p-subgroup, outer automorphism, Fitting subgroup, Periodic groups; locally finite groups, Chernikov p-groups, Subgroup theorems; subgroup growth, finite residual, Automorphism groups of groups, outer p-automorphisms, centre
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