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):...

Let G denote the projective special linear group PSL(2, q), for a prime power q. It is shown that a finite 2-subgroup of the group V(ZG) of augmentation 1 units in the integral group ring ZG of G is isomorphic to a subgroup of G. Furthermore, it is shown that a composition factor of a finite subgroup of V(ZG) is isomorphic to a subgroup of G.

In this paper‎, ‎we have shown that the coset diagrams for the‎ ‎action of a linear-fractional group $Gamma$ generated by the linear-fractional‎ ‎transformations $r:zrightarrow frac{z-1}{z}$ and $s:zrightarrow frac{-1}{2(z+1)}$ on‎ ‎the rational projective line is connected and transitive‎. ‎By using coset diagrams‎, ‎we have shown that $r^{3}=s^{4}=1$ are defining relations for $Gamma$‎. ‎Furt...

Denote by S the 2-dimensional projective special linear group PSL2(q) over the field of q elements. We determine, for all values of q > 3, the degrees of the irreducible complex characters of every group H such that S 6 H 6 Aut(S). Explicit knowledge of the character tables of PSL2(q) and PGL2(q) is used along with standard Clifford theory to obtain the degrees.

Abstract. We consider the action of the 2-dimensional projective special linear group PSL(2, q) on the projective line PG(1, q) over the finite field Fq, where q is an odd prime power. A subset S of PSL(2, q) is said to be an intersecting family if for any g1, g2 ∈ S, there exists an element x ∈ PG(1, q) such that x1 = x2 . It is known that the maximum size of an intersecting family in PSL(2, q...

let $g$ be a finite group and $pi_{e}(g)$ be the set of element orders of $g $. let $k in pi_{e}(g)$ and $s_{k}$ be the number of elements of order $k $ in $g$. set nse($g$):=${ s_{k} | k in pi_{e}(g)}$. in this paper, it is proved if $|g|=|$ pgl$_{2}(q)|$, where $q$ is odd prime power and nse$(g)= $nse$($pgl$_{2}(q))$, then $g cong $pgl$_