Relatively homogeneous locally finite permutation groups (Q1090756)

The author studies a new concept of homogeneity for permutation groups which is implicit in \textit{O. Kegel} [Rend. Semin. Mat. Univ. Padova 63, 295-300 (1980; Zbl 0454.20002)] and which combines with a concept of sharpness due to P. Neumann. Let \(G\subset Sym(X)\). G is called relatively homogeneous if for all pairs \((Y_ 1,H)\), \((Y_ 2,H)\) where H is a finitely generated subgroup of G and \(Y_ 1\), \(Y_ 2\) each consist of finitely many orbits of H on X and for every permutation isomorphism f: (Y\({}_ 1,H)\to (Y_ 2,H)\) with \(f(h)=h\) (h\(\in H)\) there exists \(g\in C_ G(H)\) which induces f. G is called sharp if every \(1\neq g\in G\) fixes only finitely many elements of X. The following are the main results. Theorem 1. There exist \(2^{\omega}\) non-isomorphic countable locally finite groups G possessing sharp, relatively homogeneous permutation representations and such that the finite subgroups of G which possess normal complements in G form a local system of G. Theorem 2. Suppose \(G\subset Sym(\omega)\) is countable, locally finite, and sharp and every non-trivial maximal p-subgroup of G is infinite for all primes p. Then \(G\subset G^*\subset Sym(\omega)\) where \(G^*\neq G\) is locally finite, sharp and relatively homogeneous. - It remains open if many isomorphism types of \(G^*\) in Theorem 2 can be constructed.