Orbit sizes and character degrees. III (Q2782016)

This is the third paper devoted to the investigation of the orbit structure of finite group actions [for part II cf. ibid. 516, 27-114 (1999; Zbl 0941.20004)]. The following theorems are proved. Theorem 2.1. Let \(G\) be a solvable group acting faithfully and irreducibly on a finite vector space \(V\). Then \(\text{dl}(G)\leq 24\cdot\log(m(G,V))+364\), where \(\text{dl}(G)\) is the derived length of \(G\) and \(m(G,V)\) is the number of nontrivial orbit sizes of \(G\) on \(V\).NEWLINENEWLINENEWLINEThis result on orbit sizes can be translated into the following Corollary. Let \(G\) be a solvable group \(G\). Then \(\text{dl}(G/F(G))\leq 24\cdot\log(|\text{cd}(G)|)+364\), where \(F(G)\) is the Fitting subgroup of \(G\).NEWLINENEWLINENEWLINEThis is an essential contribution to the so called Taketa problem: Is it true that \(\text{dl}(G)\) is logarithmically bounded in terms of \(|\text{cd}(G)|\)?NEWLINENEWLINENEWLINEWe need additional notation. If, as above, a solvable group \(G\) acts on a finite vector space (or an Abelian prime power group) \(V\) and let \(p\) be a prime, then \(b(G,V)\) is the number of different \(p\)-parts among the orbit sizes of \(G\) on \(V\). Next, setting \(G=G_0\), we define \(G_{i+1}=\Phi(G_i)\) for all \(i\geq 0\), where \(\Phi(G)\) is the Frattini subgroup of \(G\). Clearly, \(G_n=\{1\}\) for some \(n\); and the smallest such \(n\) is denoted by \(\phi(G)\). Put \(\phi(\{1\})=0\).NEWLINENEWLINENEWLINETheorem 1.1. Let \(G\) be a group and \(p\) a prime. Suppose that \(G\) acts faithfully on a finite Abelian \(p'\)-group \(V\). Suppose that \(P\) is a normal \(p\)-subgroup of \(G\). Then there exists an Abelian normal subgroup \(A\) of \(G\) with \(A\leq P\) such that for \(s=\phi(P/A)-1\) there exist \(v_i\in V\) (\(i=1,\dots,s\)) with the following properties: NEWLINE\[NEWLINE|G|_p>|\text{C}_G(v_1)|_p>\cdots>|\text{C}_G(v_s)|_p,\quad P>\text{C}_P(v_1)>\cdots>\text{C}_P(v_s).NEWLINE\]NEWLINE In particular, \(m(G,V)\geq b(G,V)\geq s\).NEWLINENEWLINENEWLINETheorem 1.1. implies that \(\text{dl}(P)\leq b(G,V)+2\) (for \(p=2\) this bound was obtained in Theorem 2.1 of [\textit{I. M. Isaacs}, J. Algebra 148, No. 1, 264-273 (1992; Zbl 0786.20004)]).NEWLINENEWLINENEWLINETheorem 1.5. Let \(G\) be a solvable group and \(p\) a prime. Suppose that \(G\) acts on a finite Abelian \(p'\)-group \(V\). Let \(P=\text{O}_p(G)\). Then \(\text{dl}(P/\text{C}_P(V))\leq 16\cdot\log_2(b(G,V))+286\). In particular, \(\text{dl}(P/\text{C}_P(V))\leq 16\cdot\log_2(m(G,V))+286\).NEWLINENEWLINENEWLINETheorem 2.3. Let \(G\) be a solvable group and \(p\) a prime. Let \(\text{cd}_p(G)\) denote the set of the degrees of the irreducible Brauer characters at the prime \(p\). Put \(c=|\text{cd}_p(G)|\). Then \(\text{dl}(G/\text{O}_p(G))\leq 3(c-1)(16\cdot\log_2c+286)+1\).NEWLINENEWLINENEWLINEIn the proofs, some deep results of finite \(p\)-group theory are used.