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$a$ and $b$ are elements of a group $G$, with the property that $ab=ba$.

Problems:

a) Demonstrate that if $a$ and $b$ have infinite order, then $ab$ may have either finite or infinite order.

b) If $a$ has finite order and $b$ has infinite order, then determine and prove whether $ab$ has infinite or finite order.

c) If $a$ and $b$ have finite order, show with proof whether $ab$ has finite or infinite order.

exp ikx
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lj_growl
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    Since you're new, you may not know it, but you should provide your attemp or, at least, your thoughts – Harnak Jan 30 '19 at 19:07
  • Oh, I see! For part a) I was looking at the integers closed under addition. With a=4 and b=-4. However, I am not seeing how this works for a*b. For b) I was thinking that the answer was infinite order and proof by contradiction. However, I'm not sure how this would work. For c, I wrote a proof and assumed a and b had finite order and showed that their product would be (ab)^n=e and therefore, ab had finite order! – lj_growl Jan 30 '19 at 19:09
  • For a, consider that * is the group operation, in your case $+$, so it's just $a*b = 4 - 4$. Good work with c! :) – Harnak Jan 30 '19 at 19:18
  • Thank you! I'm happy to know I was on the right track with those. For b, Could I start letting a have finite order and b infinite? So, a^n=e. Then assume to the contrary that (ab)^n=e and then show that this contradicts the assumption that b has infinite order as b is written to the power (mn)? – lj_growl Jan 30 '19 at 19:27

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