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:


1. Do you have a reference (if possible self contained, but that's probably too much to ask)?
2. 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



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


1. $A\_8 \cong {\rm PSL}\_4(2)$ and ${\rm PSL}\_3(4)$ of order 20160.
2. 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.


