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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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