# Abelian Subgroups

• October 7th 2011, 09:17 PM
veronicak5678
Abelian Subgroups
Find a non-abelian group whose proper subsets are all abelian.
• October 7th 2011, 09:30 PM
Deveno
Re: Abelian Subgroups
which groups have you tried?
• October 7th 2011, 09:50 PM
veronicak5678
Re: Abelian Subgroups
S_3 , GLn
• October 7th 2011, 11:57 PM
Deveno
Re: Abelian Subgroups

i presume you mean "subgroups" instead of "subsets", because "abelian set" makes no sense at all....
• October 8th 2011, 05:00 AM
Swlabr
Re: Abelian Subgroups
Quote:

Originally Posted by veronicak5678
S_3 , GLn

HINT: Try $S_3$ again...
• October 8th 2011, 05:10 AM
Deveno
Re: Abelian Subgroups
aw, i was rooting for you suggesting GL2(F2)....
• October 8th 2011, 05:57 AM
Swlabr
Re: Abelian Subgroups
Quote:

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...
• October 8th 2011, 06:42 AM
Deveno
Re: Abelian Subgroups
um, did you not catch the joke i made?
• October 8th 2011, 10:56 AM
Drexel28
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, $S_3$!
• October 8th 2011, 11:56 AM
Deveno
Re: Abelian Subgroups
oh you're so cute when you're clever....
• October 8th 2011, 01:19 PM
Drexel28
Re: Abelian Subgroups
Quote:

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

(Blush) Oh stop!
• October 10th 2011, 02:24 AM
Swlabr
Re: Abelian Subgroups
Quote:

Originally Posted by Deveno
aw, i was rooting for you suggesting GL2(F2)....

Nope, sorry, still don't get it...
• October 10th 2011, 07:53 AM
Deveno
Re: Abelian Subgroups
GL2(F2) is isomorphic to S3
• October 10th 2011, 07:58 AM
Swlabr
Re: Abelian Subgroups
Quote:

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...)
• October 10th 2011, 08:21 AM
Deveno
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.