Suppose that  is a finite group. Then the set of all prime divisors of  is denoted by  and the set of element orders of  is denoted by . Suppose that . Then the number of elements of order  in  is denoted by  and the sizes of the set of elements with the same order is denoted by ; that is, . In this paper, we prove that if  is a group such that , where , then . Here  denotes the family of Suzuk...

Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

The sum of distances between all the pairs of vertices in a connected graph is known as the {it Wiener index} of the graph. In this paper, we obtain the Wiener index of edge complements of stars, complete subgraphs and cycles in $K_n$.

Let G be a simple primitive subgroup of Sn, speciied in terms of a set of generating permutations. If jGj n 5 , eecient algorithms are presented that nd \the most natural permutation representation" of G. For example, if G is a classical group then we nd a suitable projective space underlying G. A number of related questions are considered. Our notion of \eeciency" takes into account many exist...

Let G be a finite group , in this paper using the order and largest element order of G we show that every finite group with the same order and largest element order as G 2 (q), where q 11 is necessarily isomorphic to the group G 2 (q)

Proposition 1. (Grunewald [5, Proposition B]) Let F be a free group of rank k ≥ 2, generated by {s1, . . . , sk}. Let r1, . . . , rm be words over {s1, . . . , sk} ±1 and R their normal closure 〈〈r1, . . . , rm〉〉F in F . Let H be the group with presentation 〈s1, . . . , sk | r1, . . . , rm〉 and φ the canonical epimorphism φ : F → H ∼= F/R. Let F be a free group of rank k generated by {t1, . . ....

Let G be a finite group and let $GK(G)$ be the prime graph of G. We assume that $n$ is an odd number. In this paper, we show that if $GK(G)=GK(B_n(p))$, where $ngeq 9$ and $pin {3,5,7}$, then G has a unique nonabelian composition factor isomorphic to $B_n(p)$ or $C_n(p)$ . As consequences of our result, $B_n(p)$ is quasirecognizable by its spectrum and also by a new proof, the ...

Using the Luthar–Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru. As a consequence, for this group we confirm Kimmerle’s conjecture on prime graphs.