Irreducible modular characters and \(p\)-regular conjugacy classes (Q2747692)

Let \(G\) be a finite solvable group and let \(p\) be a prime divisor of \(|G|\). The authors define two subsets of prime divisors of \(|G|\) as follows: NEWLINE\[NEWLINE\rho_p'(G)=\{q\text{ a prime}:q\text{ divides }|C|\},NEWLINE\]NEWLINE where \(C\) runs over the conjugacy classes of \(p\)-regular elements of \(G\), and NEWLINE\[NEWLINE\rho_p(G)=\{r\text{ a prime}:r\text{ divides }\chi(1)\},NEWLINE\]NEWLINE where \(\chi\) runs over the irreducible \(p\)-modular Brauer characters of \(G\). The paper contains various results relating to these two subsets, of which the following may be noted.NEWLINENEWLINENEWLINETheorem 1.6. We have \(\rho_p(G)\leq\rho_p'(G)\).NEWLINENEWLINENEWLINETheorem 1.7. Let \(r\) and \(s\) be distinct primes different from \(p\) and suppose that the \(\{r,s\}\)-length of \(G\) is 1. Suppose also that there exists an irreducible \(p\)-modular Brauer character \(\chi\) of \(G\) such that \(rs\) divides \(\chi(1)\). Then there exists a conjugacy class \(C\) of \(p\)-regular elements such that \(rs\) divides \(|C|\).NEWLINENEWLINENEWLINEWe note that there is a misprint in the statement of Theorem 1.6 in the paper, where \(\rho_{p'}\) is written in place of \(\rho_p'\). Furthermore, the statement of a result of Uno, described on p. 57 of the paper, omits to say that \(|A|\) and \(|G|\) are assumed to be relatively prime. The notation used for \(\{r,s\}\)-length in Theorem 1.7 may not be standard and should preferably be defined in words. What the authors call a \(p\)-separable group is more familiarly called a \(p\)-solvable group. Finally, a number of the results in the paper are well known. For example, Lemma 2.3 is a standard result known in much greater generality.