An amalgamation for finite group extensions
The authors give a complete criterion for the existence of completions of certain types of group amalgams. The amalgams considered are of the type (*): \(K\leftarrow B\to H\) where K, B and H are finite extensions of a fixed group G. A completion of (*) is a group J generated by K and H; thus J is an extension of G. The question is ``under what conditions is it possible to produce a completion J of (*) with the property that J/G is finite?'' The main result gives necessary sufficient conditions for the existence of such a completion. This is the content of Theorem 1: Let G be a group and (*) be an amalgamation as described above: then there is a completion J of (*) with J/G finite if and only if \(K/C_ K(G)\) and \(H/C_ H(G)\) generate a finite subgroup of the outer automorphism group of G. The methods as well as the problem were motivated by the work of \textit{K. Hickin} [Arch. Math. 45, 485-491 (1985; Zbl 0581.20026)] and \textit{K. Hickin} and \textit{R. E. Phillips} [Group Theory, Proc. 2nd Int. Conf., Bressanone/Italy 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23, 143- 171 (1990)].
scientific article; zbMATH DE number 4160999
scientific article