The group of central automorphisms of the standard wreath products (Q1057364)

The structure of the group \(Aut_ c(W)\) of central automorphisms of the standard wreath product \(W=A Wr B\) of two groups A and B is studied. A necessary and sufficient condition is given under which the group of central automorphisms coincides with the group I(W) of the inner automorphisms of W. Let \(K_ c=Aut_ c(W)\cap K\) and \(I_ c=Aut_ c(W)\cap I_ 1\), where K is the group of all automorphisms of \(W=A Wr B\) which leave B elementwise fixed and \(I_ 1\) is the group of those inner automorphisms of W induced by the elements of the subgroup \(A^ B\) of W. The following three theorems are proved. Theorem 1. Let \(W=A Wr B\) where if B is of order 2 then A is not of order 2 or A is not a dihedral group of order \(4m+2\). Then \(Aut_ c(W)=K_ c\times I_ c\). - Theorem 2. Let \(W=A Wr B\). (i) If A is not of exponent 2 when \(| B| =2\), then \(Aut_ c(W)=K_ c\times I_ c\) with \(I_ c=Z_ 2(W)/Z(W)\). (ii) If A is of exponent 2, \(A\neq C_ 2\) and \(| B| =2\), then \(Aut_ c(W)=K_ c\times I_ 1\). - Theorem 3. Let \(W=A Wr B\) with A and B nontrivial. Then \(Aut_ c(W)=I(W)\) if and only if \(A=B=C_ 2\).