Uni Stuff > Mathematics
Abstract Algebra
humph:
--- Quote from: /0 on July 15, 2010, 05:59:01 pm ---Thanks humph, I might return to that problem a bit later though...
I thought my Analysis professor did a lot of proof copying... but then, perhaps I'm just not used to that style of teaching. Analysis proofs give me headaches.
--- End quote ---
Analysis proofs do indeed induce headaches. Not sure about proof copying, a lot of the time there's only so many ways you can prove some statement, so it's not so much stealing as proving it the natural way...
--- Quote from: /0 on July 15, 2010, 05:59:01 pm ---Anyway...
With questions like:
"Prove that the inverse of an element in a subgroup is the same as the inverse of that element in "
Are the proofs meant to be as trivial as this:
Suppose . Let and .
Then since , too.
So you have ... and by taking inverses
--- End quote ---
Yeah, that's pretty much it. You should mention that the identity in a subgroup is the same as that in the group, and the last step isn't taking inverses, but simply by the uniqueness of inverses.
--- Quote from: /0 on July 15, 2010, 05:59:01 pm ---"Show by example that the product of elements of finite order in a nonabelian group need not have finite order."
Hmmm... I thought for a while permutations groups might work... but now I'm sure they won't, since for permutations and , there exists a permutation , which has finite order.
Pretty much the only other non-abelian group I know is ... probably not worth it?
--- End quote ---
It's clearly not going to work with permutation groups, because they have finite order, so every element in them must have finite order. So you'll need some sort of nonabelian infinite group. Matrix groups are really the obvious examples, unfortunately - you probably want , as everything interesting that happens with nonabelian infinite groups should happen in this group. Mind you, it could take a bit of effort to find the example that you want (and I really can't be bothered...).
/0:
Thanks again humph
(I didn't mean 'copying' like that, I guess what I meant was we could just as well get a lot of proofs from the book, but examples would be helpful)
TI-89 to the rescue!
,
humph:
--- Quote from: /0 on July 15, 2010, 07:08:34 pm ---Thanks again humph
(I didn't mean 'copying' like that, I guess what I meant was we could just as well get a lot of proofs from the book, but examples would be helpful)
TI-89 to the rescue!
,
--- End quote ---
How do you actually prove the last statement? (Hint: look up nilpotent matrices. This is covered very briefly in MATH1115.)
/0:
I guess I would proceed by induction.
It appears that
Where are Fibonacci numbers
Then I would use induction
I'm not sure how you would use nilpotent matrices however... since none of the matrices here seem to be nilpotent.
zzdfa:
Another way you could prove that is to write and then use the binomial theorem to expand that.
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