arXiv subjects
Publications and source records attributed to María José Felipe.
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We report on recent results concerning minimal $G$-invariant characters of $N$ (which are the sums of the characters on each orbit of the action of $G$ by conjugation on $\text{Irr}(N)$) and their influence on the structure of $N$, as well as their relationship to the $G$-conjugacy classes of $N$.
It is well known that finite groups with exactly two character degrees have an abelian derived subgroup and, consequently, are solvable. Let $G$ be a finite group and $N$ a normal subgroup of $G$. In this paper, we prove that normal subgroups possessing exactly two degrees of minimal $G$-invariant characters are solvable. Furthermore, it is shown that if these degrees are $\{1, f\}$ for some integer $f$, then either $f$ is a prime power or the commutator subgroup $[N,G]$ is abelian. Whether $[N, G]$ is abelian when $f$ is a prime power remains an open problem. Specifically, we prove that this holds when $f=p$.
We study the conditions under which the head characters of a finite solvable group, as defined by I. M. Isaacs, behave well with respect to restriction. We also determine the intersection of the kernels of all head characters of the group. Using G. Navarro's definition of $\mathfrak{F}'$-characters, we generalize these results for any saturated formation $\mathfrak{F}$ containing the formation of nilpotent groups.
Let $p$ be a prime. In this paper we classify the $p$-structure of those finite $p$-separable groups such that, given any three non-central conjugacy classes of $p$-regular elements, two of them necessarily have coprime lengths.
A theorem of Z. Arad and E. Fisman establishes that if $A$ and $B$ are two conjugacy classes of a finite group $G$ such that either $AB=A\cup B$ or $AB=A^{-1} \cup B$, then $G$ cannot be non-abelian simple. We demonstrate that, in fact, $\langle A\rangle = \langle B\rangle$ is solvable, the elements of $A$ and $B$ are $p$-elements for some prime $p$, and $\langle A\rangle $ is $p$-nilpotent. Moreover, under the second assumption, it turns out that $A=B$ and this is the only possible case. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.
Let $N$ be a normal subgroup of a finite group $G$. From a result due to Brauer, it can be derived that the character table of $G$ contains square submatrices which are induced by the $G$-conjugacy classes of elements in $N$ and the $G$-orbits of irreducible characters of $N$. In the present paper, we provide an alternative approach to this fact through the structure of the group algebra. We also show that such matrices are non-singular and become a useful tool to obtain information of $N$ from the character table of $G$.
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs $Γ_G(N)$ and $Γ^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes of these classes, respectively. These graphs are subgraphs of the ordinary ones associated to the conjugacy classes ofG, $Γ(G)$ and $Γ^{\ast}(G)$, which have been widely studied by several authors. We prove that the number of connected components of both graphs is at most 2, we determine the best upper bounds for the diameters and characterize the structure of $N$ when these graphs are disconnected.
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the graph $Γ_G(N)$, which is the graph associated to the conjugacy classes of $G$ contained in $N$, has no triangles and when the graph consists in exactly one triangle.
We prove that if a finite group $G$ contains a conjugacy class $K$ whose square is of the form $1 \cup D$, where $D$ is a conjugacy class of $G$, then $\langle K \rangle$ is a solvable proper normal subgroup of $G$ and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all non-central conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.
Suppose that $G$ is a finite group and $K$ a non-trivial conjugacy class of $G$ such that $KK^{-1}=1\cup D\cup D^{-1}$ with $D$ a conjugacy class of $G$. We prove that $G$ is not a non-abelian simple group. We also give arithmetical conditions on the class sizes determining the structure of $\langle K\rangle$ and $\langle D\rangle$. Furthermore, if $D=K$ is a non-real class, then $\langle K\rangle$ is $p$-elementary abelian for some odd prime $p$.
Many results have been established that show how the number of conjugacy classes appearing in the product of classes affect the structure of a finite group. The aim of this paper is to show several results about solvability concerning the case in which the power of a conjugacy class is a union of one or two conjugacy classes. Moreover, we show that the above conditions can be determined through the character table of the group.
Let $G$ be a finite group and $N$ a normal subgroup of $G$. We determine the structure of $N$ when the diameter of the graph associated to the $G$-conjugacy classes contained in $N$ is as large as possible, that is, is equal to three.
Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there exists only a finite number of finite groups $G$, up to isomorphism, having a normal subgroup $N$ of index $n$ which contains exactly $s$ non-central $G$-conjugacy classes. We provide upper bounds for the orders of $G$ and $N$, which are used by using GAP to classify all finite groups with normal subgroups having a small index and few $G$-classes. We also study the corresponding problems when we only take into account the set of $G$-classes of prime-power order elements contained in a normal subgroup.
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that have only been partially solved.
We give a characterization of the finite groups having nilpotent or abelian Hall $π$-subgroups which can easily be verified from the character table.