Non-isomorphic finite simple groups

2012-09-19T20:43:20Z

Hello,

The smallest integer $n$ such that there exists two non-isomorphic simple groups of order $n$, is $n=20160$ (namely for the groups $\mathrm{PSL}_3(\mathbb F _4)$ and $\mathrm{PSL}_4(\mathbb F _2)$). I read that there are infinitely many integer $n$ such that here exists two non-isomorphic simple groups of order $n$. I have two questions:

Do you have a reference (if possible self contained, but that's probably too much to ask)?
I suspect that it is "rare" to find such an integer. For instance if we denote by $a_k$ the orders of non-cyclic simple groups ($a_1=60$, $a_2=168$, $a_3=360$,....) and $b_k$ the integers such that there exists two non-isomorphic simple groups of order $b_k$, then I guess that $\displaystyle \lim\frac{b_k}{a_k}=+\infty$. Do you know if this is the case?

Thanks

Answer by Derek Holt for Non-isomorphic finite simple groups

2012-09-20T08:25:55Z

Just to summarise the comments: the only nonisomorphic finite simple groups with the same orders are

$A_8 \cong {\rm PSL}_4(2)$ and ${\rm PSL}_3(4)$ of order 20160.
The groups ${\rm P \Omega}_{2n+1}(q)$ and ${\rm PSp}_{2n}(q)$ for all odd prime powers $q$ and $n \ge 3$. These have order

$$(q^{n^2} \Pi_{i=1}^n (q^{2i}-1))/2$$

For references, see Gerry Myerson's comment.

Non-isomorphic finite simple groups - MathOverflow

Non-isomorphic finite simple groups

Answer by Derek Holt for Non-isomorphic finite simple groups