On multiply transitive locally finite permutation groups (Q1111694)

By using techniques developed in the theory of multiply transitive finite permutation groups, several theorems on multiply transitive locally finite permutation groups are obtained. We assume that the set of positive integers is a subset of an infinite set \(\Omega\). First the following two theorems are proved. Theorem 1. Let G be an eight-transitive locally finite permutation group on an infinite set \(\Omega\). Then \(G_{1,2,...,8}\) has a nonidentity 2-subgroup P which is not semiregular on \(\Omega\)-I(P). - Theorem 2. Let p be an odd prime and G be a \((p^ 2+p)\)-transitive locally finite permutation group on an infinite set \(\Omega\). Then \(G_{1,2,...,p^ 2+p}\) has a nonidentity p- subgroup P which is not semiregular on \(\Omega\)-I(P). - As a corollary to Theorems 1 and 2 the following theorem is proved. Theorem 3. Let p be an odd prime and G be a t-transitive locally finite permutation group with \(t=8\) or \(p^ 2+p\) on an infinite set \(\Omega\). Then G has a nonidentity element fixing at least 2t points of \(\Omega\). In 1985 Nagao proved that if G is a 4-transitive permutation group on \(\{\) 1,2,...,n\(\}\) with \(G\neq S_ 5\) (the symmetric group of degree 5), \(A_ 6\) (the alternating group of degree 6), \(M_{11}\) (the Mathieu group of degree 11), then \(I(G_{1,2,3,4})=\{1,2,3,4\}\) holds. Last the following two theorems are proved. Theorem 4. Let G be a t-transitive locally finite permutation group with \(t\geq 8\) on an infinite set \(\Omega\). If \(G_{1,2,...,t}\) has a finite maximal 2-subgroup (\(\neq 1)\), then \(I(G_{1,2,...,t})=\{1,2,...,t\}\) holds. - Theorem 5. Let p be an odd prime and G be a t-transitive locally finite permutation group with \(t\geq p^ 2+p\) on an infinite set \(\Omega\). If \(G_{1,2,...,t}\) has a finite maximal p-subgroup (\(\neq 1)\), then \(I(G_{1,2,...,t})=\{1,2,...,t\}\) holds.