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In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two non-isomorphic simple groups of order $n$? The two easiest families of finite simple groups - $\mathbb{Z}_p$ for $p$ a prime and $A_n$ for $n\ge5$ - clearly don't yield any examples.

It should be possible to check this using the classification of finite simple groups, but I don't know enough about the fabled 'sixteen infinite families of groups of Lie type' to do the check myself. Also, if the answer is 'no', I'd be much more interested to see an elementary(ish) proof than I would be to see a proof that relies on over 3,000 pages of non-stop hard group theory.

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This has been asked before over on MO and answered affirmatively: mathoverflow.net/questions/107620/… –  yatima2975 Jun 6 '13 at 14:04
    
Ah, OK. Thanks for the link. –  Donkey_2009 Jun 6 '13 at 14:06
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I think this will help: math.stackexchange.com/questions/1423/… –  Keneth Adrian Jun 6 '13 at 14:09

1 Answer 1

up vote 4 down vote accepted

The links in the comments already give very good answers, but here is something similar:

There are very few coincidences amongst the orders of the various finite simple groups in the various families. Many of these coincidences are explained by exceptional isomorphisms for groups that have more than one characteristic:$ \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\PSU}{\operatorname{PSU}} \newcommand{\PSp}{\operatorname{PSp}} \newcommand{\PSO}{\operatorname{P\Omega}} $

  • Order 60, $A_5 \cong \PSL(2,4) \cong \PSL(2,5)$
  • Order 168, $\PSL(2,7) \cong \PSL(3,2)$
  • Order 360, $A_6 \cong \PSL(2,9)$
  • Order 20160, $A_8 \cong \PSL(4,2)$
  • Order 25920, $\PSO(5,3) \cong \PSU(4,2)$

Also one has various low dimension isomorphisms in general:

  • $\PSU(2,q) \cong \PSL(2,q)$
  • $\PSO(5,q) \cong \PSp(4,q)$

But also a slightly weird one that is only an isomorphism in characteristic 2:

  • $\PSO(2n+1,2^f) \cong \PSp(2n,2^f)$

This slightly weird one forces the orders of $\PSO(2n+1,q)$ and $\PSp(2n,q)$ to be the same for all $q$, however. So we get the order coincidence:

  • $|\PSO(2n+1,q)| = |\PSp(2n,q)|$, but $\PSO(2n+1,q) \not\cong\PSp(2n,q)$ for odd $q$

The only other order coincidence is the very special $A_8 \cong \PSL(4,2)$ versus $\PSL(3,4)$ of order 20160.

These results are proved for the classical and exceptional groups in Artin (1955). The original observation of the two distinct simple groups of order 20160 is from Schottenfels (1899). While Artin's technique work for all groups of Lie type (the twisted types are no hard), it might be nice to see a post CFSG version in Garge (2005), which also handles direct products of simple groups and knows the orders of all sporadic simple groups.

  • Schottenfels, Ida May. “Two non-isomorphic simple groups of the same order 20,160.” Ann. of Math. (2) 1 (1899/00), no. 1-4, 147–152. MR1502265 DOI:10.2307/1967281

  • Artin, Emil. “The orders of the classical simple groups.” Comm. Pure Appl. Math. 8 (1955), 455–472. MR73601 DOI:10.1002/cpa.3160080403

  • Garge, Shripad M. “On the orders of finite semisimple groups.” Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 4, 411–427. MR2184201 DOI:10.1007/BF02829803

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Thanks very much. This is a great answer, and it's really interesting. –  Donkey_2009 Jun 6 '13 at 19:39

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