A note on barely transitive permutation groups satisfying min-\(2\)
A permutation group \(G\) on an infinite set \(\Omega\) is barely transitive if it is transitive but for every proper subgroup \(K\) of \(G\), all the \(K\)-orbits on \(\Omega\) are finite. In this paper the author investigates locally finite barely transitive groups \(G\) with point stabilizer \(H\). It is shown that if \(G\) has an involution and \(H\) satisfies min-2, then \(G\cong C_{2^ \infty}\). (This was proved by the author in an earlier paper with any prime \(p\) in place of 2, but using the classification of finite simple groups; the present proof does not, but uses the Feit- Thompson Theorem). Other results in the paper include the following: (1) If for some prime \(p\), \(H\) has min-\(p\), then so does \(G\). (2) If for some prime \(p\), every \(p\)-subgroup of \(H\) is soluble, then \(G\) is a union of a chain of proper normal subgroups. (3) If the order of every simple section of \(H\) is bounded, then the conclusion of (2) holds. (4) If there are \(Y \triangleleft X < G\) with \(X/Y\) an infinite simple group, then \(Y\) is infinite and not locally soluble, and \(H\) involves an isomorphic copy of \(X/Y\).
scientific article; zbMATH DE number 510093