arXiv subjects
Publications and source records attributed to Juan Martínez Madrid.
A new type of conjectures on characters of finite groups, related to the McKay conjecture, have recently been proposed. In this paper, we study these conjectures for symmetric groups.
Given a group $G$ we write $h(G)$ to denote the maximum number of times that a field extension of $\mathbb{Q}$ appears as the field of values of a conjugacy class of a group. In this work, we prove that $|G|$ is bounded in terms of $h(G)$. Moreover, we classify the groups with $h(G)\leq 3$.
Let $G = X \wr H$ be the wreath product of a nontrivial finite group $X$ with $k$ conjugacy classes and a transitive permutation group $H$ of degree $n$ acting on the set of $n$ direct factors of $X^n$. If $H$ is semiprimitive, then $k(G) \leq k^n$ for every sufficiently large $n$ or $k$. This result solves a case of the non-coprime $k(GV)$ problem and provides an affirmative answer to a question of Garzoni and Gill for semiprimitive permutation groups. The proof does not require the classification of finite simple groups.
We classify all finite groups such that all irreducible character degrees appear with multiplicity at most $2$. As a consequence, we prove that the largest group with at most $2$ irreducible characters of the same degree is the Baby Monster.
Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.