arXiv subjects

Kukil Kalpa Rajkhowa

Publications and source records attributed to Kukil Kalpa Rajkhowa.

4 recordsLinked to original sources

arXiv2026

Group action-stabilizer graph of group actions of a group on a set

In this paper we introduce the group action-stabilizer graph $GAS(G)$ of a group $G$ on a set $X$ with vertex set as the collection of all the group actions of $G$ on $X$, and any two vertices $ϕ$ and $ψ$ are adjacent if and only if the non-trivial subgroups $\cap G_x^ϕ$ and $\cap G_x^ψ$ of $G$ intersect non-trivially, where $G_x^ϕ$ and $G_x^ψ$ are two stabilizers of $x$ with respect to the actions $ϕ$ and $ψ$, respectively. We characterize a special subgraph $gas(G)$ of $GAS(G)$ in which the vertex set contains the actions $ϕ$ of $G$ on $X$ such that $\cap G_x^ϕ$'s are distinct. We determine the number of group actions within some specific groups and find certain conditions under which $GAS(G)$ is equal to $gas(G)$. We also examine the conditions under which $gas(G)$ and its complement are derived graph for a finite nilpotent group $G$.

math.GR↗

arXiv2024

Some results on GVZ-groups with two character degrees

We investigate the finite groups $G$ for which $χ(1)^{2}=|G:Z(χ)|$ for all characters $χ\in Irr(G)$ and $|cd(G)|=2$, where $cd(G)=\{χ(1)| χ\in Irr(G)\}$. We call such a group a GVZ-group with two character degrees. We establish bijections between the sets of characters of some groups obtained from a GVZ-group with two character degrees. Additionally we obtain some alternate characterizations of a GVZ-group with two character degrees and we construct a GVZ-group having the character degree set $\{1,p\}$.

math.GR↗

arXiv2024

GVZ-groups with two character degrees

We investigate the finite groups $G$ for which $χ(1)^{2}=|G:Z(χ)|$ for all characters $χ\in Irr(G)$ and $|cd(G)|=2$. We obtain some alternate characterizations of these groups and we obtain some information regarding the structure of these groups.

math.GR↗

arXiv2024

$p$-groups with small number of character degrees and their normal subgroups

If $G$ be a finite $p$-group and $χ$ is a non-linear irreducible character of $G$, then $χ(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fernández-Alcober and Moretó obtained the relation between the character degree set of a finite $p$-group $G$ and its normal subgroups depending on whether $|G/Z(G)|$ is a square or not. In this paper we investigate the finite $p$-group $G$ where for any normal subgroup $N$ of $G$ with $G'\not \leq N$ either $N\leq Z(G)$ or $|NZ(G)/Z(G)|\leq p$ and obtain some alternate characterizations of such groups. We find that if $G$ is a finite $p$-group with $|G/Z(G)|=p^{2n+1}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $N\leq Z(G)$, then $cd(G)=\{1, p^{n}\}$. We also find that if $G$ is a finite $p$-group with nilpotency class not equal to $3$ and $|G/Z(G)|=p^{2n}$ and $G$ satisfies the condition that for any normal subgroup $N$ of $G$ either $G'\not \leq N$ or $|NZ(G)/Z(G)|\leq p$, then $cd(G) \subseteq \{1, p^{n-1}, p^{n}\}$.

math.GR↗