Related papers: Finite groups in which every proper characteristic…

We classify the finite groups $G$ which satisfies the condition that every complex irreducible character,whose degree's square doesn't divide the index of its kernel in $G$, lies in the same Galois conjugacy class.

In a series of recent contributions on the notion of global breadth $\mathbf{B}(G)$ of a finite group $G$, it was interesting to observe the structural conditions arising from the classification of finite groups of $\mathbf{B}(G)=8$. This…

Brou\'e, Malle and Rouquier conjectured in that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for…

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of…

We consider a finiteness condition on centralizers in a group G, namely that |C_G (x) : <x>| is finite for every non-normal cyclic subgroup <x> of G. For periodic groups, this is the same as |C_G (x)| is finite for every non-normal cyclic…

A family of groups is called (maximal) cyclic bounded ((M)CB) if, for every natural number $n$, there are only finitely many groups in the family with at most $n$ (maximal) cyclic subgroups. We prove that the family of groups of prime power…

Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any $b\in G$, the subgroup \( \langle a, b \rangle \) is cyclic. The \emph{non-cyclic graph} $\Gamma(G)$ of \( G…

A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by…

A finite group $G$ is called uniformly semi-rational if there exists an integer $r$ such that the generators of every cyclic sugroup $\langle x \rangle$ of $G$ lie in at most two conjugacy classes, namely $x^G$ or $(x^r)^G$. In this paper,…

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in $[0,\frac{1}{3}]$, answering a question of Cameron, and that the number of those…

A finite group $G$ is said to be a $\mathcal{B}_{\psi}$-group if $\psi(H)<|G|$ for any proper subgroup $H$ of $G$, where $\psi(H)$ denotes the sum of element orders of $H$. In this paper, we characterize the $\mathcal{B}_{\psi}$-groups up…

We classify finite groups with a small average number of zeros in the character table.

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.

A skew-morphism of a finite group $G$ is a permutation $\s$ on $G$ fixing the identity element, and for which there exists an integer function $\pi$ on $G$ such that $\s(xy)=\s(x)\s^{\pi(x)}(y)$ for all $x,y\in G$. It has been known that…

A supercharacter theory for a finite group $G$ is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility…

Let $G$ be a finite group. We will say that $M$ and $S$ form a \textsl{complete splitting} (\textsl{splitting}) of $G$ if every element (nonzero element) $g$ of $G$ has a unique representation of the form $g=ms$ with $m\in M$ and $s\in S$,…

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its…

We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm…

For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified…

A group $G$ is said to have dense normalizers if each non-empty open interval in its subgroup lattice $L(G)$ contains the normalizer of a certain subgroup of $G$. In this note, we find all finite groups satisfying this property. We also…