On \(C\)-automorphisms of finite groups admitting a strongly 2-embedded subgroup (Q2834190)

Let \(U(\mathbb Z G)\) be the unit group of the integral group ring \(\mathbb Z G\), where \(G\) is a finite group. Suppose, that \( N_{U(\mathbb ZG)}(G)\) is the normalizer of \(G\) in \(U(\mathbb ZG)\), \(Z (U(\mathbb ZG))\) is the center of \(U(\mathbb ZG)\) and Inn\((G)\) is the group of the inner automorphisms of \(G\). It is said that for \(G\) holds the normalizer problem, if \( N_{U(\mathbb ZG)}(G) = G. Z (U(\mathbb ZG))\). An automorphism \(\sigma\) of \(G\) is called a \(C\)-automorphism if (I) the restriction of \(\sigma\) to each Sylow subgroup of \(G\) equals the restriction of some inner automorphism of \(G\); (II) \(\sigma\) is a class-preserving automorphism; (III) \({\sigma}^2\in\mathrm{Inn}(G)\) (see \textit{Z.S. Marciniak} and \textit{K.W. Roggenkamp} [Roggenkamp, Klaus W. (ed.) et al., Algebra -- representation theory. Proceedings of the NATO Advanced Study Institute, Constanta, Romania, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 28, 159--188 (2001; Zbl 0989.20002)].NEWLINENEWLINE Many authors study \(C\)-automorphisms. If \(\mathrm{Aut}_C(G)\) is the group of all \(C\)-automorphisms of \(G\), then the authors set Out\(_C(G)=\mathrm{Aut}_C(G)/\mathrm{Inn}(G)\). A subgroup \(M\) of the group \(G\) is said to be a strongly 2-embedded subgroup in \(G\) if (i) \(M < G\) and 2 divides \(| M| \); (ii) If \(g \in G\backslash M\), then 2 does not divide \(| M \cap M^g| \). For the characterization of the groups with a strongly 2-embedded subgroup see \textit{H. Bender} [J. Algebra 17, 527--554 (1971; Zbl 0237.20014)].NEWLINENEWLINE If \(G\) is of even order and the intersection of two different Sylow 2-subgroups is 1, then \(G\) is called a (TI)-group, see [\textit{M. Suzuki}, Ann. Math. (2) 80, 58--77 (1964; Zbl 0122.03202)].NEWLINENEWLINE The main results of this paper are the following.NEWLINENEWLINE Theorem A. Let \(G\) be a group with a strongly 2-embedded subgroup. Then \(\mathrm{Out}_C(G)\) = 1. In particular, the normalizer property holds for \(G\).NEWLINENEWLINE Corollary B. Let \(G\) be a (TI)-group. Then \(\mathrm{Out}_C(G)\) = 1. In particular, the normalizer property holds for \(G\).