Re: How many countable groups are there?

In article <We2ne.817$wy1.697@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
W. Dale Hall <mailtodhall@xxxxxxxxx> wrote:

>> Don't know off-hand if there would be an easier way of "exhibiting" c
>> groups which are patently non-isomorphic...
>>
>
> From a self-professed know-nothing:
>
>Given a subset K of the prime integers, form the direct sum
>
> S_K = \bigoplus_{k \in K} ( Z/kZ )
>
>What seems apparent (to a know-nothing) is that S_K will have torsion
>for those primes in K and no others. Thus, if K,L are distinct subsets
>of the primes, S_K and S_L are non-isomorphic.
>
>If K is infinite, then S_K is countable.
>
>I expect something's wrong with this example, since it seems too
>straightforward.

Nah, it's fine. I just have a fondness for using cannonballs to swat
flies.

Note, of course, that "countable" normally includes finite as well as
countably infinite...

--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.