# Proving a property when elements of a group commute

## Main Question or Discussion Point

By commutative, we know that ##ab = ba## for all a,b in G. Thus, why do we need to prove separately that ##a^n b^m = b^ma^n##? Isn't it the case that ##a^n## and ##b^m## are in fact elements of the group? So shouldn't the fact that they commute automatically be implied?

## Answers and Replies

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blue_leaf77
Science Advisor
Homework Helper
Isn't it the case that anana^n and bmbmb^m are in fact elements of the group?
Yes but each of ##a^n## and ##b^m## might equal another element of the group and they need not commute if the group is non-Abelian.

Yes but each of ##a^n## and ##b^m## might equal another element of the group and they need not commute if the group is non-Abelian.
Oh, I see. I neglected to see that the group is not actually abelian

mathman
Science Advisor
The proof you want seems trivial.

WWGD
Science Advisor
Gold Member
2019 Award
By commutative, we know that ##ab = ba## for all a,b in G. Thus, why do we need to prove separately that ##a^n b^m = b^ma^n##? Isn't it the case that ##a^n## and ##b^m## are in fact elements of the group? So shouldn't the fact that they commute automatically be implied?
Why do you believe you need to prove it separately?

lavinia
Science Advisor
Gold Member
By commutative, we know that ##ab = ba## for all a,b in G. Thus, why do we need to prove separately that ##a^n b^m = b^ma^n##? Isn't it the case that ##a^n## and ##b^m## are in fact elements of the group? So shouldn't the fact that they commute automatically be implied?
If you know that the group is commutative then there is nothing to prove. ##a^nb^m = b^ma^n## by definition. If you do not know that the group is commutative then you need a proof.

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mathman
Science Advisor
Proof: $$a^nb^m=a^{n-1}bab^{m-1}$$, etc.