Kevin Wampler wrote:
> Orchid XP v8 wrote:
>> I was under the impression the result applies to all groups - which is 
>> why the groups of prime order are always cyclic.
> 
> It is true that every subgroup of a finite group has an order which 
> divides that of the group (this result has actually been around longer 
> than group theory itself).  This implies that all prime order groups are 
> cyclic.
> 
> The converse, however, does not hold.  Some finite groups do not have 
> any subgroups corresponding to some of their order's divisors.

Ah, right. I guess I missed that technicallity.

Of course, intuitively, it totally makes sense that the order of a 
subgroup would have to be a factor of the group's order. I just thought 
that every group could be split into subgroups. So I guess there are 
large non-prime groups that don't have any [nontrivial] subgroups then?

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

Post a reply to this message