Wreath products and enlargements of groups
Let \(N^*_ G(\Delta) = \bigcap_{g\in G} g^{-1}\Delta g\) the autonormalizer of \(\Delta\) in a group \(G\). The subgroup \(\Delta\) is coreless in \(G\) if \(N^*_ G(\Delta) = 1\). The group \(G\) called an enlargement of a \(\theta\) group \(H\) by a group \(J\) if there exist an exact sequence \(1 \to H {\overset\lambda\longrightarrow} G \to J \to 1\) and a normal subgroup \(\Delta\) of \(\lambda(H)\) such that \(\lambda(H)/\Delta \approx \Theta\) and is coreless in \(G\). In this paper the author investigates the relationship between group enlargements and wreath products. By the Extension Lemma of Suzuki (Let \(H\) and \(J\) be any two groups such that \(H\) is centreless. Then every outer action of \(J\) on \(H\) is uniquely realized by an extension of \(H\) by \(J\)) and the Schreier Conjecture (Every finite simple group has a solvable outer automorphism group) he gets the following consequences: The direct product \(H\times J\) is the only extension of a nonabelian simple group \(H\) by a nonabelian simple group \(J\).
scientific article; zbMATH DE number 429130
scientific article