Non-isomorphic finite simple groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:11:06Z http://mathoverflow.net/feeds/question/107620 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107620/non-isomorphic-finite-simple-groups Non-isomorphic finite simple groups Portland 2012-09-19T20:43:20Z 2012-09-20T08:25:55Z <p>Hello,</p> <p>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:</p> <ol> <li>Do you have a reference (if possible self contained, but that's probably too much to ask)?</li> <li>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?</li> </ol> <p>Thanks</p> http://mathoverflow.net/questions/107620/non-isomorphic-finite-simple-groups/107660#107660 Answer by Derek Holt for Non-isomorphic finite simple groups Derek Holt 2012-09-20T08:25:55Z 2012-09-20T08:25:55Z <p>Just to summarise the comments: the only nonisomorphic finite simple groups with the same orders are</p> <ol> <li><p>$A_8 \cong {\rm PSL}_4(2)$ and ${\rm PSL}_3(4)$ of order 20160.</p></li> <li><p>The groups <code>${\rm P \Omega}_{2n+1}(q)$</code> and ${\rm PSp}_{2n}(q)$ for all odd prime powers $q$ and $n \ge 3$. These have order</p></li> </ol> <p>$$(q^{n^2} \Pi_{i=1}^n (q^{2i}-1))/2$$</p> <p>For references, see Gerry Myerson's comment.</p>