Zhang, Jiping Complex linear groups of degree at most \(| P| -2\). (English) Zbl 0737.20007 Southeast Asian Bull. Math. 15, No. 1, 87-91 (1991). 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. Reviewer: M.E.Harris (Minneapolis) MSC: 20D05 Finite simple groups and their classification 20C15 Ordinary representations and characters 20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20G40 Linear algebraic groups over finite fields Keywords:finite group; TI subgroup; TI abelian Sylow \(p\)-subgroup; degree; simple group; faithful irreducible complex character PDFBibTeX XMLCite \textit{J. Zhang}, Southeast Asian Bull. Math. 15, No. 1, 87--91 (1991; Zbl 0737.20007)