McKay natural correspondences on characters

Character correspondeces in solvable groups with a self-normalizing Sylow subgroup

“…It is worth mentioning that, already in 2003, Navarro suspected that the p-solvability condition above could be replaced by the condition that p is odd. This prediction was confirmed in [NTV14], where the authors also provided a new method to prove Theorem 2.1. The following extension result is the key of the new method.…”

Sylow branching coefficients for symmetric groups

“…For example, in [14] it is shown that is normal in if and only if all irreducible constituents of have degree coprime to . At the opposite end of the spectrum, in [17] it is shown that when is odd, then the Sylow ‐subgroup is self‐normalising if and only if is the only constituent of of degree coprime to .…”

“…For instance, Problem 12 of Brauer's article [1] and the famous Brauer Height Zero Conjecture [1,Problem 23] ask what amount of the algebraic structure of a Sylow p-subgroup can be read off the character table of a finite group. More recently, it has been noted that given a finite group G with Sylow p-subgroup P , the permutation character 1 P ↑ G controls important structural properties of the entire group G. For example, in [14] it is shown that P is normal in G if and only if all irreducible constituents of 1 P ↑ G have degree coprime to p. At the opposite end of the spectrum, in [17] it is shown that when p is odd, then the Sylow p-subgroup P is self-normalising if and only if 1 G is the only constituent of 1 P ↑ G of degree coprime to p.…”

Characters of odd degree of symmetric groups

“…Sometimes, we do not only have that , but it is also possible to determine a natural bijection between and , where is the set of complex irreducible characters of of degree not divisible by . This occurs for solvable groups and (see ), for groups with a normal ‐complement, using the Glauberman correspondence and for groups having self‐normalizing Sylow ‐subgroup for odd (see ).…”