arXiv subjects
Publications and source records attributed to Juan Martínez.
Let $G_p$ be the set of $p$-elements of a finite group $G$. Do we need all the Sylow $p$-subgroups of $G$ to cover $G_p$? Although this question does not have an affirmative answer in general, our work indicates that the answer is yes more often than one could perhaps expect.
Let $G$ be a finite group, let $π$ be a set of primes and let $p$ be the smallest prime in $π$. In this work, we prove that $G$ possesses a normal and abelian Hall $π$-subgroup if and only if the probability that two random $π$-elements of $G$ commute is larger than $\frac{p^2+p-1}{p^3}$. We also prove that if $x$ is a $π$-element not lying in $O_π(G)$, then the proportion of $π$-elements commuting with $x$ is at most $1/p$.
Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $|G|$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is nilpotent or supersolvable.
In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.