# For which numbers there is only one simple group of that order?

There is only one simple group of orders: 3, 60, and 360 respectivley. Are there other groups of this kind? What general characteritics do they share? From pure curiosity did this question arise. Thanks for any help.

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All other primes are also obvious examples. – Chris Eagle Jun 3 '12 at 16:25
Well, indeed... I think I should change the question to mean...simple nonabelian group of that order, and then take off the example of n=3, right? – awllower Jun 3 '12 at 16:34
See this question. – Mikko Korhonen Jun 3 '12 at 17:04
Ah! Sorry, this seems to be a silly question... – awllower Jun 3 '12 at 17:06

The orders of all finite simple groups have been worked out as falling into certain infinite families (e.g. the cyclic groups of prime order) or certain 26 sporadic groups.

According to the Wikipedia article listing finite simple groups, there are very limited orders where two simple groups share that order:

"In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group $B_n(q)$ has the same order as $C_n(q)$ for q odd, n > 2; and the groups $A_8 = A_3(2)$ and $A_2(4)$ both have orders 20160."

Added: The classification of finite simple groups is a major milestone of group theory, one whose proof is spread across dozens of papers and thousands of pages, mostly published in the years 1955-1983. After the orders of these were known, noting which orders appear more than once in the list of groups was comparatively trivial. No order appears more than twice.

A group is simple iff it has no normal subgroup other than the identity and itself. It is easy to show the commutator subgroup is always normal, so for a simple group $G$ the commutator subgroup is either the identity (which means G is abelian) or the group G itself (which means G is perfect).

The above definition of simple group suggests a parallel with prime numbers. For finite abelian simple groups the parallel is exact. Since any subgroup of an abelian group is normal, a finite abelian group is simple if and only if its order is prime, i.e. the cyclic group $\mathbb{Z}_p$ for $p$ prime.

The first big piece of the classification of finite simple groups is the Odd Order Thm. by Feit and Thompson (1963). It is usually stated as all odd order finite groups are solvable, but an equivalent statement would be that any odd order finite simple group is abelian, i.e. cyclic of odd prime order.

It follows that those orders which admit two distinct simple groups will be even. Using the notation from the earlier link, these orders can be explicitly stated:

• The alternating group $A_8$ is isomorphic to $A_3(2)$ but distinct from $A_2(4)$, with which it shares order $\frac{8!}{2} = 20160$.

• The simple groups $B_n(q)$ and $C_n(q)$ for q an odd prime power, n > 2, are distinct but have the same order:

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

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I think I need more time to digest some information... – awllower Jun 3 '12 at 17:15