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I was thinking about how to predict the order of an element a*b if the orders of a and b are known where a and b are elements of some group.

One textbook I have gives the result (without proof) that if a and b commute and their orders are relatively prime, then order(a*b) = order(a)*order(b). But I have been unable to prove this result. Can someone help me out with this and explain if there is any result if a and b are not relatively prime.

Also, to me it seems logical that nothing specific can be said about the order of a*b if a and b do not commute. Can someone please tell me if I am right or correct me if not.

TIA

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# Question on order of an element

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