# Math Help - Abelian Subgroups

1. ## Abelian Subgroups

Find a non-abelian group whose proper subsets are all abelian.

2. ## Re: Abelian Subgroups

which groups have you tried?

S_3 , GLn

4. ## Re: Abelian Subgroups

and what are your conclusions?

i presume you mean "subgroups" instead of "subsets", because "abelian set" makes no sense at all....

5. ## Re: Abelian Subgroups

Originally Posted by veronicak5678
S_3 , GLn
HINT: Try $S_3$ again...

6. ## Re: Abelian Subgroups

aw, i was rooting for you suggesting GL2(F2)....

7. ## Re: Abelian Subgroups

Originally Posted by Deveno
aw, i was rooting for you suggesting GL2(F2)....
Well, I would have suggested a Tarskii Monster Group, but I thought that it was perhaps overkill...

8. ## Re: Abelian Subgroups

um, did you not catch the joke i made?

9. ## Re: Abelian Subgroups

Just a remark, $S_3$ is definitely a good choice, but why? It's a first-week-of-group-theory matter that every group of order at most five is abelian, thus if you can find a non-abelian group whose proper subgroups all have size at most five, then you're golden. But, clearly any group of order six has the property that every proper subgroup has order at most five, and so in particular, any non-abelian group of order six will have the property you seek. Ta-da, $S_3$!

10. ## Re: Abelian Subgroups

oh you're so cute when you're clever....

11. ## Re: Abelian Subgroups

Originally Posted by Deveno
oh you're so cute when you're clever....
Oh stop!

12. ## Re: Abelian Subgroups

Originally Posted by Deveno
aw, i was rooting for you suggesting GL2(F2)....
Nope, sorry, still don't get it...

13. ## Re: Abelian Subgroups

GL2(F2) is isomorphic to S3

14. ## Re: Abelian Subgroups

Originally Posted by Deveno
GL2(F2) is isomorphic to S3
Oh. Apparently I cannot count - I thought it had 8 elements! (I quickly counted matrices with zero determinant - too quickly...)

15. ## Re: Abelian Subgroups

that's ok, i once proved $\mathbb{N}^{\mathbb{N}}$ was countable, which if correct, would surely have gained me a fields medal.