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Coleman automorphisms of standard wreath products of nilpotent groups by groups with prescribed Sylow 2-subgroups. (English) Zbl 1303.16038
Let \(G\) be a finite group, \(\mathbb ZG\) its integral group ring. The authors, in a series of papers, a recent one is [Commun. Algebra 42, No. 6, 2502-2509 (2014; Zbl 1298.16022)], consider the normalizer problem that is, whether in \(U(\mathbb ZG)\) the normalizer \(N_{U(\mathbb ZG)}(G)\) is \(GZ(U(\mathbb ZG))\). Denote by \(\mathrm{Out}_{\mathbb Z}(G)\) the factor of the group of automorphisms of \(G\) induced by conjugation by some element in \(N_{U(\mathbb ZG)}(G)\) by the group of inner automorphisms. It is clear that the normalizer property for \(G\) is equivalent to triviality of \(\mathrm{Out}_{\mathbb Z}(G)\). A Coleman automorphism of \(G\) is a class-preserving automorphism with inner square and with action restricted to any Sylow-subgroup of \(G\) coinciding with that of some inner automorphism. Denote by \(\mathrm{Out}_C(G)\) the factor of the group of Coleman automorphisms of \(G\) by the group of inner automorphisms. It is clear that triviality of \(\mathrm{Out}_C(G)\) implies the normalizer property for \(G\). In the past decade in this direction there was intensive research performed, recent papers include M. Hertweck and E. Jespers [J. Group Theory 12, No. 1, 157-169 (2009; Zbl 1168.16017)] and the authors [Publ. Math. 82, No. 3-4, 599-605 (2013; Zbl 1274.20021)].
The main result of the paper is that if \(G=N\text{\,wr\,}H\) is the wreath product of the nilpotent group \(N\) by \(H\) which has either cyclic, dihedral or generalized quaternion 2-Sylow subgroup then \(\mathrm{Out}_C(G)\) is trivial. This also yields that if \(G=N\text{\,wr\,}H\) is the wreath product of the Abelian group \(N\) by the Frobenius group \(H\) then \(\mathrm{Out}_C(G)\) is trivial.

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20D45 Automorphisms of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20E22 Extensions, wreath products, and other compositions of groups
Full Text: DOI
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