On the finite group with a T. I. Sylow \(p\)-subgroup (Q1175592)

A subgroup \(X\) of a finite group \(G\) is said to be a T.I. subgroup if whenever \(g\in G\) and \(X\cap X^ g\neq 1\), one has \(X=X^ g\). The main results of this paper are: Theorem 4: Let \(G\) be a finite group with a T.I. Sylow \(p\)-subgroup \(P\). If \(p=3\) or 5, assume that \(G\) contains no composition factor isomorphic to the simple groups \(SL(2,8)\) or \(Sz(2^ 5)\). If \(G\) has a normal subgroup \(W\) such that \(p|(| W|,| G/W|)\), then \(G\) is \(p\)-solvable. Theorems 8 and 10: Let \(G\) be a finite group with a non-normal T.I. Sylow \(p\)-subgroup \(P\) and \(p>11\). Then: (a) \(P\) is cyclic if and only if \(G\) contains no composition factors isomorphic to \(PSL(2,p^ n)\) with \(n>1\) or \(PSU(3,p^ m)\) with \(m\geq 1\); and (b) \(P\) is abelian if and only if \(G\) has no composition factors isomorphic to \(PSU(3,p^ m)\) with \(m\geq1\).