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The prime graph of finite group $G$, is as follows: the vertex set is prime divisor of $|G|$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $pq$. Let $G$ be a simple group such that $|G|\mid q(q^{2}-1)/2$ where $q=p^{n}$, $p$ is prime ($p\mid |G|$). Let the number of connected components of prime graph $G$ are at least $2$ ( $p$ is an isolated vertex).

My question: Is there any classification for such group $G$?

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  • $\begingroup$ I am a bit puzzled. Any integer $m$ divides $q-1$ for a suitable $q=p^m$. So, the class of simple groups $G$ such that $Card(G)$ divides $q(q^2-1)/2$ is the class of all simple groups. You do not assume that $p$ divides the order of $G$, so can this question be phrased purely in terms of a simple group $G$? $\endgroup$
    – Venkataramana
    Commented Apr 6, 2013 at 4:18

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This article should give you what you're looking for along with orders for the groups listed.

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