Related papers: Maximal representation dimension of finite p-group…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The…

Every finite group $G$ has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length $\lambda (G)$ as the minimum number of nonsoluble factors in a series of…

After Landau, we define g(n) as the maximal order of a permutation of the symmetric group S(n) on n letters. We give several estimates of the largest prime divisor of g(n).

We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.

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…

We determine an upper bound for the size of a stem $p$-group in terms of its breadth and its maximum character degree.

The aim of the present paper is to obtain a classification of all the irreducible modular representations of the symmetric group on $n$ letters of dimension at most $n^3$, including dimension formulae. This is achieved by improving an idea,…

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…

Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $\chi(1)$ for some $\chi \in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k…

For a finite group $G$, let $p(G)$ denote the minimal degree of a faithful permutation representation of $G$. The minimal degree of a faithful representation of $G$ by quasi-permutation matrices over the fields $\mathbb{C}$ and $\mathbb{Q}$…

The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable…

We present a proof of an upper bound for the lengths of finite dimensional representations of algebras obeying a modified PBW property, including Lie algebras and quantum groups. The sharpness of the bound is proved and discussed.

Given an irreducible representation of $SL_2(F_q)$ for an odd prime $q\geq 5$, we find the dimension of the space of cusp forms with respect to the full modular group taking values in the representation space. The dimension equals the…

In the paper we study finitely generated linear groups of finite rank which have faithful irreducible primitive representations over a field of characteristic zero. We prove that if an infinite finitely generated linear group $G$ of finite…

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an…

We explore the concept of additive diameters in the context of group representations, unifying various noncommutative Waring-type problems. Given a finite-dimensional representation $\rho \colon G \to \mathrm{GL}(V)$ and a subspace $U \leq…

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $\Omega(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1…

Given a representation of a finite group $G$ over some commutative base ring $\mathbf{k}$, the cofixed space is the largest quotient of the representation on which the group acts trivially. If $G$ acts by $\mathbf{k}$-algebra automorphisms,…

We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of…