Related papers: Groups of type L_2(q) acting on polytopes

We classify embeddings of the finite groups $A_4$, $S_4$ and $A_5$ in the Lie group $G_2(\mathbb{C})$ up to conjugation.

Let K/F be a cyclic extension of prime degree l over a number field F. If F has class number coprime to l, we study the structure of the l-Sylow subgroup of the class group of K. In particular, when F contains the l-th roots of unity, we…

Let $L(x)$ be any $q$-linearized polynomial with coefficients in $\mathbb{F}_q$, of degree $q^n$. We consider the Galois group of $L(x)+tx$ over $\mathbb{F}_q(t)$, where $t$ is transcendental over $\mathbb{F}_q$. We prove that when $n$ is a…

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic…

We construct four infinite families of chiral $3$-polytopes of type $\{4, 8\}$, with $1024m^4$, $2048m^4$, $4096m^4$ and $8192m^4$ automorphisms for every positive integer $m$, respectively. The automorphism groups of these polytopes are…

It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G…

This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.

Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G=\boldsymbol{G}^{\sigma}$ its fixed-point subgroup under the action of a Steinberg endomorphism $\sigma$, and $\overline{G}$ an almost simple group…

If $G$ is a transitive group of degree $n$ having a string C-group of rank $r\geq (n+3)/2$, then $G$ is necessarily the symmetric group $S_n$. We prove that if $n$ is large enough, up to isomorphism and duality, the number of string…

In this short note we prove that the finite non-abelian simple groups PSL(2,q), where q = 5,7, are determined by their posets of classes of isomorphic subgroups. In particular, this disproves the conjecture in the end of [5].

Let $\L_g^G$ denote the locus of hyperelliptic curves of genus $g$ whose automorphism group contains a subgroup isomorphic to $G$. We study spaces $\L_g^G$ for $G \iso \Z_n, \Z_2{\o}\Z_n, \Z_2{\o}A_4$, or $SL_2(3)$. We show that for $G \iso…

In this article we show that if ${\cal V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,...,c_s),\ {\mathcal N}_{c_1,c_2,...,c_s}$, and $G\cong{\bf {Z}}_{p^{\alpha_1}}\stackrel{n}{*}{\bf…

The type-PQ adjacency polytope associated to a simple graph is a $0/1$-polytope containing valuable information about an underlying power network. Chen and the first author have recently demonstrated that, when the underlying graph $G$ is…

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The…

Let $m$ be a positive integer such that $p$ does not divide $m$ where $p$ is prime. In this paper we find the number of conjugacy classes of completely reducible cyclic subgroups in GL$(2, q)$ of order $m$, where $q$ is a power of $p$.

We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among…

Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of…

Let $\F_q$ be the finite field with $q$ elements, $p=\Char \F_q$. The group $\GL_2(\F_q)$ acts naturally in the set of irreducible polynomials over $\F_q$ of degree at least $2$. In this paper we are interested in the characterization and…

Let $\mathcal{G}_{\alpha}(X, G)$ be the $G$-gauge group over a space $X$ corresponding to a map $\alpha \colon X \to BG$. We compute the integral cohomology of $B\mathcal{G}_{1}(S^2, SO(n))$ for $n = 3,4$. We also show that the homology of…

This paper constitutes a first step in the author's program to investigate the question of when a homotopy of 2-cocycles $\omega = \{\omega_t\}_{t \in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of…