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ON p-SOLVABILITY AND AVERAGE CHARACTER DEGREE IN A FINITE GROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 27 July 2023, pp. 507-511
- Print publication:
- June 2024
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- Article
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Assume that G is a finite group, N is a nontrivial normal subgroup of G and p is an odd prime. Let
$\mathrm{Irr}_p(G)=\{\chi \in \mathrm{Irr}(G) : \chi (1)=1~\mathrm{or}~ p \mid \chi (1)\}$ and 
$\mathrm{Irr}_p(G|N)=\{\chi \in \mathrm{Irr}_p(G) : N \not \leq \mathrm{ker}\,\chi \}$. The average character degree of irreducible characters of 
$\mathrm{Irr}_p(G)$ and the average character degree of irreducible characters of 
$\mathrm{Irr}_p(G|N)$ are denoted by 
$\mathrm{acd}_p(G)$ and 
$\mathrm{acd}_p(G|N)$, respectively. We show that if 
$\mathrm{Irr}_p(G|N) \neq \emptyset $ and 
$\mathrm{acd}_p(G|N) < \mathrm{acd}_p(\mathrm{PSL}_2(p))$, then G is p-solvable and 
$O^{p'}(G)$ is solvable. We find examples that make this bound best possible. Moreover, we see that if 
$\mathrm{Irr}_p(G|N) = \emptyset $, then N is p-solvable and 
$P \cap N$ and 
$PN/N$ are abelian for every 
$P \in \mathrm{Syl}_p(G)$.
