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?
Anyone know anything about Abstract Algebra?
Anyone know anything about Abstract Algebra?
What part of lockbox do you not understand!?
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Re: Anyone know anything about Abstract Algebra?
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.
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.
Re: Anyone know anything about Abstract Algebra?
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.Edgecrusher wrote: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.
What part of lockbox do you not understand!?
Re: Anyone know anything about Abstract Algebra?
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
But I'm sure of one thing, the world is flat.
Sorry I can't help
Re: Anyone know anything about Abstract Algebra?
Swag,
You are so Cro-Mag. The world is round and the universe revolves around it.
You are so Cro-Mag. The world is round and the universe revolves around it.
Re: Anyone know anything about Abstract Algebra?
I'm with Swag on this one, if you keep walking eventually your fall off.
The truth is there, just don't look blindly
Re: Anyone know anything about Abstract Algebra?
Yes, there's plenty of cliffs around to fall off of ;)
Re: Anyone know anything about Abstract Algebra?
algebra is just a figment of your imagination
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