Complex linear groups of degree at most \(| P| -2\) (Q1175613)

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 result of this paper is: Theorem A: Let \(G\) be a finite group with a T.I. abelian Sylow \(p\)-subgroup \(P\) of order \(| P|>7\). If \(G\) possesses a faithful complex character \(\chi\) of degree \(\chi(1)\leq| P|-3\), then either \(P\trianglelefteq G\) or \(G'/O_ p(G')\cong PSL(2,| P|)\). It is well-known that the simple group \(SL(2,8)\) has an irreducible complex character of degree 7 and a cyclic T.I. Sylow 3-subgroup of order 9. Therefore the bound \(| P|-3\) for the degree \(\chi(1)\) is best possible. Also by mimicing the proof of Theorem A, the author obtains: Theorem B: Let \(G\) be a finite group with a T.I. abelian Sylow \(p\)-subgroup of order \(| P|>p\). If \(G\) has a faithful irreducible complex character \(\chi\) of degree \(| P|-2\), then either \(P\trianglelefteq G\) or \(| P|=3^ 2\) and \(G\cong Z(G)\times SL(2,8)\). The proofs of these results naturally divide into the cases: (1) \(G\) is \(p\)-solvable and (2) \(G\) is not \(p\)-solvable.