IA-automorphisms of permutational wreath products (Q2773009)

An automorphism of a group \(G\) is called an IA-automorphism if it induces the identity on the factor commutator group \(G/G'\). If the group of all the IA-automorphisms of \(G\) coincides with the group of inner automorphisms of \(G\), then \(G\) is called semicomplete (see [\textit{S. Andreadakis}, J. Lond. Math. Soc. 44, 361-364 (1969; Zbl 0164.33801)]). In the paper under review, the author continues his study to find necessary and sufficient conditions for a permutational wreath product to be semicomplete (see [\textit{J. Panagopoulos}, Algebra Colloq. 7, No. 3, 275-280 (2000; Zbl 0976.20020)]). The problem of semicompleteness was fully answered in the special case of the standard wreath products (see [\textit{J. Panagopoulos}, Arch. Math. 37, 499-511 (1981; Zbl 0458.20027)]). Here the author gives some necessary conditions for the semicompleteness of the permutational wreath product \(W\) of two groups \(A\) and \(B\). In the case of finite groups \(A\) and \(B\) (in which the unrestricted and restricted permutational wreath products of \(A\) and \(B\) are the same) where \(A\) is Abelian, the author gives necessary and sufficient conditions under which the group \(W\) is semicomplete, i.e. in the main result of the paper (Theorem 5.8) it is proved:NEWLINENEWLINENEWLINELet \(A\) and \(B\) be finite groups and \(H\triangleleft B\). Suppose that \(W=A\wr_HB\), \(A\) is Abelian and \(1\neq Z(B)\not\leq H\). Then \(W\) is semicomplete if and only if \(W=C_2\wr_HB\), where \(B\) is semicomplete, \(|Z(B)|=2\), and \(\Hom(H,C_2)=1\), or \(W=C_2\wr_HB\), where \(B\) is semicomplete and \(|Z(B)|=3\), or \(W=C_3\wr_HB\), where \(B\) is semicomplete and \(|Z(B)|=2\).NEWLINENEWLINENEWLINENote that the results of this paper are obtained under the global hypothesis that the base groups of the restricted permutational products are characteristic subgroups.