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Carmen Melchor

Publications and source records attributed to Carmen Melchor.

9 recordsLinked to original sources
arXiv2024

An Arad and Fisman's theorem on products of conjugacy classes revisited

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.

math.GR↗
arXiv2024

Graphs associated to conjugacy classes of normal subgroups in finite groups

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.

math.GR↗
arXiv2024

Triangles in the graph of conjugacy classes of normal subgroups

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.

math.GR↗
arXiv2024

Squares of real conjugacy classes in finite groups

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.

math.GR↗
arXiv2024

Multiplying a conjugacy class by its inverse in a finite group

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 $p$.

math.GR↗
arXiv2024

Powers of conjugacy classes in a finite group

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.

math.GR↗
arXiv2024

Landau's Theorem on conjugacy classes for normal subgroups

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.

math.GR↗
arXiv2024

Some problems about products of conjugacy classes in finite groups

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.

math.GR↗