Anyone know anything about Abstract Algebra?

[Hobbs]

So, I'm taking a group theory class, and we're using Herstein's book. The problems my teacher has assigned seem to be causing me a lot of grief, and I can't seem to solve more than 2/3 of them. I know that I'm competent enough to learn this material, and I feel like the rest of the class must be struggling just as much as I am. Anyways, was wondering if there was anyone out there who could help solve this one:

Assume G is a group with H and K as subgroups. The co-set indices of H and K are both finite.
Prove that H intersect K has a co-set index that is also finite.

The proof is trivial if G is a finite group.

The trouble I'm having is showing that it's true for infinite groups G. I was wondering if anyone could help.
I have so far that if G is infinite, and H and K have finite co-set indices, then H and K have infinite order. That proof is pretty trivial, but I will enumerate it below.

Assume Not (FTSOAC)
H,K have finite co-set indicies and are finite order.
But G can be written as the union of all the cosets of H (which is a finite number). Meaning that a G is finite (because H and all the cosets of H are the same order and finite). This is contradiction, and therefore H,K must either have infinite co-set indices or infinite order.


I was thinking that from here, if you could prove that H and K were cyclic, and that they both shared an element other than the identity, then we could easily show that H is a subset of K, or vice versa, and therefore H intersect K is really just H, or K, and that the co-set index of H intersect K is then finite. I think I may be over complicating the issue though. Anyone have any input? Can you point me in the right direction? Am I totally off-kilter with my approach?

[Edgecrusher]

Group theory has been a while, so I don't have a solution ready for you, but I would say the way to prove it is by contradiction. Assume that G is infinite, then try to prove that H intersect K has an infinite coset index, given that their individual coset indices are finite. What would also help is to solve it for a specific group (say, Z) and then abstract your proof to the general case.

I assume you're talking abstract groups here, because for, say, matrix groups or topological groups the solution isn't too difficult.

If I can find time to brush up on my group theory, I'll see if I can come up with a more useful answer. Absrtract group theory is nice, but completely useless if you specialize in an applied direction.

[Hobbs]

Group theory has been a while, so I don't have a solution ready for you, but I would say the way to prove it is by contradiction. Assume that G is infinite, then try to prove that H intersect K has an infinite coset index, given that their individual coset indices are finite. What would also help is to solve it for a specific group (say, Z) and then abstract your proof to the general case.

I assume you're talking abstract groups here, because for, say, matrix groups or topological groups the solution isn't too difficult.

If I can find time to brush up on my group theory, I'll see if I can come up with a more useful answer. Absrtract group theory is nice, but completely useless if you specialize in an applied direction.

Yeah, we're discussing abstract group theory. Ironically enough, I've come to that conclusion already. I can see the way to prove it, I'm just having difficulty relating the coset index of H or K to the infinite coset index of H intersect K. I'm thinking some kind of bijection is in order, possibly from the cosets of H intersect K to the cosets of H or K, but I can't seem to make anything work.

[Swagman]

I have absolutly no idea what you 2 are talking about.

But I'm sure of one thing, the world is flat.

Sorry I can't help

[Meathead]

Swag,

You are so Cro-Mag. The world is round and the universe revolves around it.

[Blade]

I'm with Swag on this one, if you keep walking eventually your fall off.

[Page]

Yes, there's plenty of cliffs around to fall off of ;)

[Purify]

lol this is why i dropped Algebra.....

[LotuS]

algebra is just a figment of your imagination