Simple Groups from Order 2001 to Order 3640

A Simple Classification of Finite Groups of Order p2q2

‎Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, ‎respectively‎. ‎In this paper‎, ‎we show that up to isomorphism‎, ‎there are four groups of order p^2q^2 when Q and P are cyclic‎, ‎three groups when Q is a cyclic and P is an elementary ablian group‎, ‎p^2+3p/2+7 groups when Q is an elementary ablian group an...

all simple groups with order from 1 million to 5 million are efficient

‎there is much interest in finding short presentations for the finite‎ ‎simple groups‎. ‎indeed it has been suggested that all these groups are‎ ‎efficient in a technical sense‎. ‎in previous papers we produced nice‎ ‎efficient presentations for all except one of the simple groups with‎ ‎order less than one million‎. ‎here we show that all simple groups with‎ ‎order between $1$ million and $5$ ...

On Simple Groups of Finite Order. I

THE ORDER GRAPHS OF GROUPS

Let $G$ be a group. The order graph of $G$ is the (undirected)graph $Gamma(G)$,those whose vertices are non-trivial subgroups of $G$ and two distinctvertices $H$ and $K$ are adjacent if and only if either$o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate theinterplay between the group-theoretic properties of $G$ and thegraph-theoretic properties of $Gamma(G)$. For a finite group$G$, we s...

Classification of finite simple groups whose Sylow 3-subgroups are of order 9

In this paper, without using the classification of finite simple groups, we determine the structure of  finite simple groups whose Sylow 3-subgroups are of the order 9. More precisely, we classify finite simple groups whose Sylow 3-subgroups are elementary abelian of order 9.