1. ## euler func.

Prove that if a and b are elements of a finite group G, then

ord(ab) = ord(ba).

2. ## ...........

alright. and then?

3. If $\displaystyle ord(ab) = n$ then $\displaystyle (ab)^n = e\quad \Rightarrow \quad \underbrace {(ab)(ab) \cdots (ab)}_n = e$.

So $\displaystyle a\left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right]b = e\quad \Rightarrow \quad \left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right] = a^{ - 1} b^{ - 1}$

$\displaystyle (ba)^n = \underbrace {(ba)(ba) \cdots (ba)}_{n - 1}(ba) = \left( {a^{ - 1} b^{ - 1} } \right)(ba) = e$

4. Originally Posted by Plato
If $\displaystyle ord(ab) = n$ then $\displaystyle (ab)^n = e\quad \Rightarrow \quad \underbrace {(ab)(ab) \cdots (ab)}_n = e$.

So $\displaystyle a\left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right]b = e\quad \Rightarrow \quad \left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right] = a^{ - 1} b^{ - 1}$

$\displaystyle (ba)^n = \underbrace {(ba)(ba) \cdots (ba)}_{n - 1}(ba) = \left( {a^{ - 1} b^{ - 1} } \right)(ba) = e$
umm, excuse me, how are we sure that n is the least positive integer such that $\displaystyle (ba)^n = e$?

5. Originally Posted by kalagota
umm, excuse me, how are we sure that n is the least positive integer such that $\displaystyle (ba)^n = e$?
If $\displaystyle (ab)^n=e\; \Rightarrow \;(ba)^n=e$ then the same argument shows that $\displaystyle (ba)^n=e \; \Rightarrow \; (ab)^n=e$.

6. Originally Posted by Opalg
If $\displaystyle (ab)^n=e\; \Rightarrow \;(ba)^n=e$ then the same argument shows that $\displaystyle (ba)^n=e \; \Rightarrow \; (ab)^n=e$.
you don't actually answered the question.. of course i know what that means.. the question was how are we sure that n is the smallest integer that satisfies $\displaystyle (ba)^n = e$.. Ü anyways, never mind.. i almost forgot the proof (, it was a year ago!)

7. Originally Posted by kalagota
you don't actually answered the question..
Yes I have answered the question. If you know that $\displaystyle (ab)^n=e\;\Leftrightarrow \;(ba)^n=e$, it plainly follows that ab and ba have the same order.

8. Originally Posted by kalagota
you don't actually answered the question.. of course i know what that means.. the question was how are we sure that n is the smallest integer that satisfies $\displaystyle (ba)^n = e$
Look in my first reply, I deliberately leave some point for the questioner to answer.
I showed that $\displaystyle ord(ba) \leqslant n$, correct?
Now suppose that $\displaystyle ord(ba) = m$.
As Opalg pointed out the same argument shows that $\displaystyle n \leqslant m$, so $\displaystyle n = m$.

9. ## ............

what does the 'e' mean?

10. Originally Posted by anncar
what does the 'e' mean?
Identity element.

11. Originally Posted by anncar
what does the 'e' mean?
OH! My goodness!
We never know with what level we are dealing. Do we?
It is dangerous to assume anything.

12. Originally Posted by ThePerfectHacker
Identity element.
what is an identity element? Please explain

13. actually it seems self explanotory(sp?) now...kinda..but please explain..

14. alright guys. I checked on google and understand it now. nevermind..