## On outer automorphisms of Černikov p-groups.(English)Zbl 0706.20029

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 ([W. Gaschütz, J. Algebra 4, 1-2 (1966; Zbl 0142.260)], [P. Schmid, Math. Z. 147, 271-277 (1976; Zbl 0307.20016)], [F. Menegazzo and S. E. Stonehewer, J. Lond. Math. Soc., II. Ser. 31, 272-276 (1985; Zbl 0526.20023)], [R. Marconi, Rend. Semin. Mat. Univ. Padova 74, 123-127 (1985; Zbl 0588.20027)]).
Reviewer: S.Franciosi

### MSC:

 20F28 Automorphism groups of groups 20F50 Periodic groups; locally finite groups 20E07 Subgroup theorems; subgroup growth 20E36 Automorphisms of infinite groups
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### References:

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