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In this paper, we show that a locally graded group with a finite number m of non-(nilpotent of class at most n) subgroups is (soluble of class at most [log2(n)] + m + 3)-by-(finite of order $\le$ m!). Also we show that the derived length of…

We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it…

Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{\chi(1)\;|\;\chi\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, assume that $G$ has a maximal $A$-invariant subgroup $M$ that is a direct product of some isomorphic simple groups, we prove that if $G$ has a…

Let G be a finite group of order n and V an irreducible representation over the complex numbers of dimension d. For some nonnegative number e, we have n=d(d+e). If e is small, then the character of V has unusually large degree. We fix e and…

In this paper, we will show that if for every nonlinear complex irreducible character of a finite group G, some multiple of it is induced from an irreducible character of some proper subgroup of G, then G is solvable. This is a…

We prove that if the average of the degrees of the irreducible characters of a finite group $G$ is less than 16/5, then $G$ is solvable. This solves a conjecture of I.M. Isaacs, M. Loukaki, and the first author. We discuss related…

If $G$ is a finite group, an irreducible complex-valued character $\chi$ is called rational if $\chi(g)$ is rational for all $g\in G$. Also, a conjugacy class $x^G$ is called rational, if for all irreducible complex-valued character $\chi$,…

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then…

If $G$ be a finite $p$-group and $\chi$ is a non-linear irreducible character of $G$, then $\chi(1)\leq |G/Z(G)|^{\frac{1}{2}}$. In \cite{fernandez2001groups}, Fern\'{a}ndez-Alcober and Moret\'{o} obtained the relation between the character…

For every finite quasisimple group of Lie type $G$, every irreducible character $\chi$ of $G$, and every element $g$ of $G$, we give an exponential upper bound for the character ratio $|\chi(g)|/\chi(1)$ with exponent linear in $\log_{|G|}…

Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of upper triangular $n \times n$ matrices having every diagonal entry equal to $1$. It is…

Let G be a finite p-group, where p is a prime number, and $\chi$ and $\psi$ be faithful complex irreducible characters of G. We study the relation between the number $\eta(\chi,\psi)$ of distinct irreducible constituents of the product…

Using a general result of Lusztig, we find the decomposition into irreducibles of certain induced characters of the projective general linear group over a finite field of odd characteristic.

We show there exist representations of each maximal compact subgroup $K$ of the $p$-adic group $G=\mathrm{SL}(2,F)$, attached to each nilpotent coadjoint orbit, such that every irreducible representation of $G$, upon restriction to a…

A character of a finite group having degree $n$ takes values which may be expressed as sums of $n$ or fewer roots of unity. In this note, we prove a result which describes the irreducible constituents of generalized characters on abelian…

We determine the indecomposable characters of several classes of infinite dimensional groups associated with operator algebras, including the unitary groups of arbitrary unital simple AF algebras and II$_1$ factors.

Given a finite group $G$ and a prime $p$, let $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. The group $G$ satisfies the Quillen dimension property at $p$ if $\mathcal{A}_p(G)$ has non-zero homology…

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ…

We show that if G is a finite group then no chain of modular elements in its subgroup lattice L(G) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L(G) has length at least two…