Large orbits ofp-groups on characters and applications to character degrees

Fitting Height and Character Degree Graphs

“…(See [7] and the references in there for other applications of these results.) The work in [8], in the case of even order solvable groups, gives useful information only for groups of Fitting height larger than 10.…”

On p-parts of Brauer character degrees and p-regular conjugacy class sizes

“…For the solvable case, the problems in this paper have been studied in [25] and certain bounds were obtained; more explicitly, it was shown that for a finite solvable group G with O p (G) = 1, log p |G| p ≤ 96ē p (G) and log p |G| p ) ≤ 683ecl p (G). We greatly improve those bounds, and we will obtain corresponding results for arbitrary finite groups.…”

On group actions of p‐groups on p′$p^{\prime }$‐groups

“…However, for non‐abelian G not much is known about this interesting action. Moretó [7] proved an interesting result along this line. He showed the existence of a “large” orbit on $${\operatorname{Irr}}(G)$$ in case that A is a p ‐group and G is solvable such that $$(|A|,|G|)=1$$.…”

“…In the same paper, Moretó asked if one could drop the extra condition that G is solvable. In fact, he proposed the following two conjectures [7, Conjecture 2.2] .…”