Character correspondences, degrees, and derived length in solvable groups
Assume that \(A\) acts on \(G\) via automorphisms, that \((| A|,| G|)=1\), and let \(C=C_G(A)\). Let \(\text{cd}_A(G)\) be the set of degrees of \(A\)-invariant irreducible characters of \(G\) and \(\text{dl}(G)\) the derived length of a solvable group \(G\). The author shows that if \(GA\) is solvable then \(\text{dl}(C)\leq 5|\text{cd}_A(G)|\), and for groups \(G\) of odd order \(\text{dl}(C)\leq 2|\text{cd}_A(G)|\). G. Navarro has conjectured for \(G\) solvable that \(C\) has the derived length at most \(|\text{cd}_A(G)|\), and verified this conjecture for supersolvable groups. It is known that in the case \(A=\{1\}\), \(\text{dl}(G)\leq 2|\text{cd}(G)|\) [\textit{D. Gluck}, J. Lond. Math. Soc., II. Ser. 31, 457-462 (1982; Zbl 0569.20010)], and for groups of odd order \(\text{dl}(G)\leq|\text{cd}(G)|\) [\textit{T. R. Berger}, J. Algebra 39, 199-207 (1976; Zbl 0362.20012)]. In the note added in proof, the author reports that A. Turull has given examples showing that \(|\text{cd}(G)|\) is not bounded by a function of \(|\text{cd}_A(G)|\). Let \(b(G)\) be the maximal degree of irreducible characters of \(G\). Theorem 1.7. Let \(V\) be a faithful completely reducible \(G\)-module with solvable \(G\neq\{1\}\). Then one of the following occurs: (a) \(b(G)\leq{1\over 2}| V|\); (b) \(b(G)\leq| V|/24^{1/3}\) unless \(| V|=4\), \(8\), or \(9\), and \(G\) is non-Abelian of order \(6\), \(21\) or \(\text{GL}(2,3)\), respectively; (c) if \(| GV|\) is odd, then \(b(G)\leq| V|^{1/2}/3^{1/2}\). Theorem 1.9. Let \(G\) and \(V\) be such as in the previous theorem. Then one of the following occurs: (a) \(G\) has a normal subgroup \(K\) such that \(b(G/K)\leq| V|^{1/2}/24^{1/3}\); (b) \(| V|\leq 8\) and \(\text{dl}(G)\leq 2\); or (c) \(| V|=64\), \(G\) has a normal subgroup \(K\) with \(\text{dl}(K)=3b(G/K)\), and \(G/K\cong S_4\) or \(A_4\).
scientific article; zbMATH DE number 1021754
scientific article