Thread: Abelian Subgroups

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

which groups have you tried?

S_3 , GLn

and what are your conclusions?

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

Originally Posted by veronicak5678

S_3 , GLn

HINT: Try again...

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

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...

um, did you not catch the joke i made?

Just a remark, 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, !

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

Originally Posted by Deveno

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

Oh stop!

Originally Posted by Deveno

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

Nope, sorry, still don't get it...

GL2(F2) is isomorphic to S3

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...)

that's ok, i once proved was countable, which if correct, would surely have gained me a fields medal.