# How many Groups?

• Mar 20th 2006, 03:30 PM
ThePerfectHacker
How many Groups?
How many groups are there up to isomorphism?
(I know it is infinite I want to know if it is countable).

• Mar 24th 2006, 11:25 PM
rgep
No, it isn't countable. Consider the groups $\displaystyle \bigoplus_{p_i \in S}C_{p_i}$, direct products of cyclic groups, where $\displaystyle S$ runs over all possible subsets of the set of prime numbers. There are uncountably many of these groups, as the set of subsets of an infinite countable set is uncountable, and no two of them are isomorphic.
• Mar 25th 2006, 03:46 PM
ThePerfectHacker
Quote:

Originally Posted by rgep
No, it isn't countable. Consider the groups $\displaystyle \bigoplus_{p_i \in S}C_{p_i}$, direct products of cyclic groups, where $\displaystyle S$ runs over all possible subsets of the set of prime numbers. There are uncountably many of these groups, as the set of subsets of an infinite countable set is uncountable, and no two of them are isomorphic.

Actually I was thinking about the same idea :)

You are of course using finitely generated abelian groups. Then as I understand it you apply the diagnol argument to show that they are uncountable.
• Mar 26th 2006, 08:04 AM
rgep
Quote:

Originally Posted by ThePerfectHacker
You are of course using finitely generated abelian groups.

No, you need to allow arbitrary sets S. If you restrict to finite sets only for S then the resulting set of groups is countable (the set of finite subsets of a countable set is countable -- exercise).