characterization of projective general linear groups

Characterization of projective special linear groups in dimension three by their orders and degree patterns

The prime graph $Gamma(G)$ of a group $G$ is a graph with vertex set $pi(G)$, the set of primes dividing the order of $G$, and two distinct vertices $p$ and $q$ are adjacent by an edge written $psim q$ if there is an element in $G$ of order $pq$. Let $pi(G)={p_{1},p_{2},...,p_{k}}$. For $pinpi(G)$, set $deg(p):=|{q inpi(G)| psim q}|$, which is called the degree of $p$. We also set $D(G):...

Characterization of some projective special linear groups in dimension four by their orders and degree patterns

‎Let $G$ be a finite group‎. ‎The degree pattern of $G$ denoted by‎ ‎$D(G)$ is defined as follows‎: ‎If $pi(G)={p_{1},p_{2},...,p_{k}}$ such that‎ ‎$p_{1}

On the Characterization of Linear and Projective Linear Groups. I

NSE characterization of some linear groups

‎For a finite group $G$‎, ‎let $nse(G)={m_kmid kinpi_e(G)}$‎, ‎where $m_k$ is the number of elements of order $k$ in $G$‎ ‎and $pi_{e}(G)$ is the set of element orders of $G$‎. ‎In this paper‎, ‎we prove that $Gcong L_m(2)$ if and only if $pmid |G|$ and $nse(G)=nse(L_m(2))$‎, ‎where $min {n,n+1}$ and $2^n-1=p$ is a prime number.

Two-dimensional Projective Linear Groups

This article is a contribution to the study of the automorphism groups of 2 − (v, k, 1) designs. Let G act as a block-transitive and point-primitive automorphism group of a non-trivial design D with v points and blocks of size k. Set k2 = (k, v−1). Assume q = pf for some prime p and positive integer f . If q ≥ [(k2k − k2 + 1)f ]2, then Soc(G), the socle of G, is not PSL(2, q). Mathematics Subje...

Chirality and projective linear groups

In recent years the term ‘chiral’ has been used for geometric and combinatorial figures which are symmetrical by rotation but not by reflection. The correspondence of groups and polytopes is used to construct infinite series of chiral and regular polytopes whose facets or vertex-figures are chiral or regular toroidal maps. In particular, the groups PSL,(Z,) are used to construct chiral polytope...