# Abelian Subgroups

• Oct 7th 2011, 08:17 PM
veronicak5678
Abelian Subgroups
Find a non-abelian group whose proper subsets are all abelian.
• Oct 7th 2011, 08:30 PM
Deveno
Re: Abelian Subgroups
which groups have you tried?
• Oct 7th 2011, 08:50 PM
veronicak5678
Re: Abelian Subgroups
S_3 , GLn
• Oct 7th 2011, 10:57 PM
Deveno
Re: Abelian Subgroups

i presume you mean "subgroups" instead of "subsets", because "abelian set" makes no sense at all....
• Oct 8th 2011, 04:00 AM
Swlabr
Re: Abelian Subgroups
Quote:

Originally Posted by veronicak5678
S_3 , GLn

HINT: Try \$\displaystyle S_3\$ again...
• Oct 8th 2011, 04:10 AM
Deveno
Re: Abelian Subgroups
aw, i was rooting for you suggesting GL2(F2)....
• Oct 8th 2011, 04:57 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...
• Oct 8th 2011, 05:42 AM
Deveno
Re: Abelian Subgroups
um, did you not catch the joke i made?
• Oct 8th 2011, 09:56 AM
Drexel28
Re: Abelian Subgroups
Just a remark, \$\displaystyle 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, \$\displaystyle S_3\$!
• Oct 8th 2011, 10:56 AM
Deveno
Re: Abelian Subgroups
oh you're so cute when you're clever....
• Oct 8th 2011, 12:19 PM
Drexel28
Re: Abelian Subgroups
Quote:

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

(Blush) Oh stop!
• Oct 10th 2011, 01:24 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...
• Oct 10th 2011, 06:53 AM
Deveno
Re: Abelian Subgroups
GL2(F2) is isomorphic to S3
• Oct 10th 2011, 06:58 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...)
• Oct 10th 2011, 07:21 AM
Deveno
Re: Abelian Subgroups
that's ok, i once proved \$\displaystyle \mathbb{N}^{\mathbb{N}}\$ was countable, which if correct, would surely have gained me a fields medal.