Semicomplete permutational wreath products (Q2711606)

A group is called semicomplete if every automorphism which induces the identity on the factor commutator group is inner. In this paper the author studies the semicompleteness of the permutational wreath product \(W=A\text{ wr}(H)B\), \(B\) finite and \(H\) a subgroup of \(B\). The main result states that if \(W\) is semicomplete then \(A\) is semicomplete and \(N_B(H)=H\cdot Z(B)\). Here, as usual, \(N_B(H)\) is the normalizer of \(H\) in \(B\) and \(Z(B)\) the centre of \(B\). Among the corollaries is an old result of the author concerning automorphisms of standard wreath products [Arch. Math. 37, 499-511 (1981; Zbl 0458.20027)].