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# Abelian Proof Watch

1. If , prove that must be abelian.

My proof goes something like this:
, but note also that . So that

But I have a sinking feeling in my stomach that doesn't prove that is abelian. It needs to reduce down to , doesn't it?

2. (Original post by Zacken)
If , prove that must be abelian.

My proof goes something like this:
, but note also that . So that

But I have a sinking feeling in my stomach that doesn't prove that is abelian. It needs to reduce down to , doesn't it?

You're quite right: you've shown that products of squares must commute, but not that products of all elements must commute.

In fact, this kind of group is *very* specific. The first thing you should do when seeing rules like this is to check some special cases. What happens when x=y? When x=y^{-1}? When x=e?
3. (Original post by Smaug123)
You're quite right: you've shown that products of squares must commute, but not that products of all elements must commute.

In fact, this kind of group is *very* specific. The first thing you should do when seeing rules like this is to check some special cases. What happens when x=y? When x=y^{-1}? When x=e?

I'm not really seeing how that gives any intuitive help.
4. (Original post by Zacken)

I'm not really seeing how that gives any intuitive help.
Indeed, the first two do not help. But the third is a really really strong condition on : every element is self-inverse. (That is, G is a "boolean group".) Among other things, if the group is finite, it is therefore of order a power of 2.

I'm trying to think of a non-magic way to show that the group is abelian. The proof follows in one line, but the correct expression to consider is a bit unmotivated. Give me a minute
5. (Original post by Zacken)

I'm not really seeing how that gives any intuitive help.
OK, motivation: we'll go for a contradiction. Suppose . What can we deduce?
6. (Original post by Smaug123)
OK, motivation: we'll go for a contradiction. Suppose . What can we deduce?
We can deduce that ?
7. (Original post by Zacken)
We can deduce that ?
And given that are self-inverse…?
8. (Original post by Smaug123)
And given that are self-inverse…?
Then ?
9. (Original post by Zacken)
Then ?

More neatly stated (without contradiction): .
10. (Original post by Smaug123)

More neatly stated (without contradiction): .
Oh my god! I'm so dense. That was so obvious. Thank you!
11. (Original post by Zacken)
Oh my god! I'm so dense. That was so obvious. Thank you!
No problem - it's really simple if you spot the right expression to consider, but that takes a bit of intuition. Contradiction often reduces the amount of intuition you need, but tends to produce a less neat proof.
12. (Original post by Smaug123)
No problem - it's really simple if you spot the right expression to consider, but that takes a bit of intuition. Contradiction often reduces the amount of intuition you need, but tends to produce a less neat proof.
Wait a second, re-reading your proof, I can't see why . We're given that but unless we assume G is abelian?
13. (Original post by Zacken)
Wait a second, re-reading your proof, I can't see why . We're given that but unless we assume G is abelian?
, and we've already shown that for all , .
14. (Original post by Smaug123)
, and we've already shown that for all , .
Fair enough but where does the equality of come from?
15. (Original post by Zacken)
Fair enough but where does the equality of come from?
Let . Then by the original condition in the question (letting ), so . That is, .
16. (Original post by Smaug123)
Let . Then by the original condition in the question (letting ), so . That is, .
*wipes sweat off forehead* This abstractness is hard to swallow. Thanks again, it makes a load more sense now.

+Rep
17. (Original post by Zacken)
*wipes sweat off forehead* This abstractness is hard to swallow. Thanks again, it makes a load more sense now.

+Rep
Thanks sorry, do tell me if I'm being too abstract. It takes a bit of time to get the right frame of mind to just swap variables in and out for each other like that. A big part of becoming a mathematician is, in my opinion, gaining the ability to detach your understanding of a thing (eg. the element as a member of ) from its name ("", which is not helpful when it comes to deriving the fact that ).
18. (Original post by Smaug123)
Thanks sorry, do tell me if I'm being too abstract. It takes a bit of time to get the right frame of mind to just swap variables in and out for each other like that. A big part of becoming a mathematician is, in my opinion, gaining the ability to detach your understanding of a thing (eg. the element as a member of ) from its name ("", which is not helpful when it comes to deriving the fact that ).
No, no, you make way more sense than my textbook! - quite a good teacher, really.

I'm still struggling to get used to that, trying to detach my pre-conceived notions, but I always get scared that detaching them might lead to mistakes somewhere, so I juggle the two in my head. I'll get the hang of it someday.

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Updated: January 18, 2015
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