Prove: $o(ab)=o(ba)$

Question

Let $G$ be a finite group prove that $o(ab)=o(ba)$

I have started with let $(ab)^n=e$ now because $G$ is a group there is $(ab)^{-1}\in G$ but can I conclude it is $b^{-1}a^{-1}$?

Well, what is $a b b^{-1} a^{-1}=$? – Zoran Loncarevic Dec 20 '16 at 12:52 — Zoran Loncarevic, Dec 20 '16 at 12:52

score 10 · Accepted Answer · answered Dec 20 '16 at 12:52

10

Observe that $(ba)^{n+1}=b(ab)^na$.

Hence, if $(ab)^n=e$, then $(ba)^{n+1}=ba$, and therefore $(ba)^n=e$.

Now its your turn to produce a clean proof !

answered Dec 20 '16 at 12:52

Fred

77,394

why $(ba)^{n+1}=b(ab)^na$? – gbox Dec 20 '16 at 12:55
@gbox For $n = 1$ you have e.g. $(ba)^2 = (ba)(ba) = b(ab)a$ – Dominik Dec 20 '16 at 12:56
@Dominik using associativity? – gbox Dec 20 '16 at 12:57
1

@gbox Yes $ $ $ $ $ $ – Dominik Dec 20 '16 at 12:57
@ gbox: for $(ba)^{n+1}=b(ab)^na$ use an easy induction argument ! – Fred Dec 20 '16 at 13:02

score 1 · Answer 2 · answered Dec 20 '16 at 12:52

1

Yes you can, in any group $(ab)^{-1} = b^{-1}a^{-1}$, just because \begin{align*} (ab)^{-1} &= (ab)^{-1}aa^{-1}\\ &= (ab)^{-1}abb^{-1}a^{-1}\\ &= b^{-1}a^{-1} \end{align*}

answered Dec 20 '16 at 12:52

martini

84,101

Prove: $o(ab)=o(ba)$

2 Answers2

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