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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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[2] P. Schmid , Normal p-subgroups in the group of automorphisms of a finite p-group , Math. Z. , 147 ( 1976 ), pp. 271 - 277 . Article | MR 419602 | Zbl 0315.20020 · Zbl 0315.20020
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