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On class-preserving Coleman automorphisms of finite separable groups. (English) Zbl 1302.20028
One says that the normaliser problem has a positive answer for a finite group \(G\) if the only invertible elements of the integral group ring \(\mathbb ZG\) that normalise the group \(G\) are the obvious ones, i.e. the elements that belong to \(GZ(\mathcal U(\mathbb ZG))\), where \(Z(\mathcal U(\mathbb ZG))\) denotes the centre of the unit group \(\mathcal U(\mathbb ZG)\) of \(\mathbb ZG\). Equivalently, the group \(\text{Out}_{\mathbb Z}(G)=\operatorname{Aut}_{\mathbb Z}(G)/\text{Inn}(G)\) is trivial, where \(\operatorname{Aut}_{\mathbb Z}(G)\) is the subgroup of the automorphism group \(\operatorname{Aut}(G)\) consisting of those automorphisms of \(G\) that are induced by an inner automorphism of \(\mathcal U(\mathbb ZG)\). A result due to Krempa says that \(\text{Out}_{\mathbb Z}(G)\) is an elementary Abelian \(2\)-group. As shown by M. Hertweck [Ann. Math. (2) 154, No. 1, 115-138 (2001; Zbl 0990.20002)], there exist finite groups for which the normaliser property has a negative answer.
In this context, the following two groups play a crucial role: \(\text{Out}_c(G)=\operatorname{Aut}_c(G)/\text{Inn}(G)\) and \(\text{Out}_{\text{Col}}(G)=\operatorname{Aut}_{\text{Col}}(G)/\text{Inn}(G)\), where \(\operatorname{Aut}_c(G)\) consists of the (conjugacy) class-preserving automorphisms of the group \(G\) and \(\operatorname{Aut}_{\text{Col}}(G)\) consists of those automorphisms of \(G\) whose restriction to each Sylow subgroup of \(G\) equals the restriction to some inner automorphism of \(G\) (the so called Coleman automorphisms of \(G\)). It is well known that \(\text{Out}_{\mathbb Z}(G)\subseteq\text{Out}_c(G)\cap\text{Out}_{\text{Col}}(G)\). So, the normaliser problem has a positive answer for \(G\) in case the latter intersection is an odd group.
In this paper the following two results are proved for a finite group \(G\).
Theorem A: Assume \(G=NA\), where \(N\) is a nilpotent normal subgroup of \(G\) and \(A\) is an Abelian subgroup of \(G\). If the Sylow \(2\)-subgroup of \(N\) is cyclic, then \(\text{Out}_c(G)\cap\text{Out}_{\text{Col}}(G)\) is of odd order.
Theorem B: Assume \(G=A\rtimes P\), a semidirect product of an Abelian group \(A\) and a dihedral or generalised quaternion \(2\)-group \(P\). If the Sylow \(2\)-subgroup of \(A\) is cyclic, then \(\text{Out}_c(G)\cap\text{Out}_{\text{Col}}(G)\) is of odd order.
An example of Z. S. Marciniak and K. W. Roggenkamp, [in: Algebra – representation theory. Proceedings of the NATO Advanced Study Institute, Constanta, Romania, August 2–12, 2000. NATO Sci. Ser. II, Math. Phys. Chem. 28, 159-188 (2001; Zbl 0989.20002)] shows that if \(G=NA\), with \(N\) a nilpotent normal subgroup of \(G\) and \(A\) an Abelian subgroup of \(G\), then in general it is not the case that \(\text{Out}_c(G)\cap\text{Out}_{\text{Col}}(G)\) is of odd order.

MSC:
20D45 Automorphisms of abstract finite groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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References:
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