Complete finite semidirect products and wreath products.
A group \(G\) is `complete' if \(Z(G)=\{1\}\) and \(\Aut(G)=\mathrm{Inn}(G)\). This paper uses group cohomology to give a sufficient condition for a finite semidirect product \(G=N\rtimes H\) with \(C_G(N)\leq N\) to be complete and proves a partial converse. Using these results the authors fully characterize complete permutational wreath products proving the following result (Theorem 2.3): Let \(G\), \(H\) be non-trivial finite groups and let \(H\) be a permutation group acting faithfully on a finite set \(\Omega\). Let \(W=G\wr H\) and let \(B\) represent the base of \(W\). Then \(W\) is complete if and only if (a) \(G\) is complete; (b) the base \(B\) is characteristic in \(W\); (c) the image of \(H\) is self-normalizing in \(\mathrm{Out}(B)\).
scientific article; zbMATH DE number 5902951