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Let $a,b,c \in G$ and $G$ is a group, prove that the following items have the same order...

  1. $a$ and $a^{-1}$
  2. $ab$ and $ba$
  3. $abc$ and $bca$

For the first, I see that I have to operate $a^{-1}$ n times to convert $\underbrace{a*\cdots *a}_n = e$ in $e$ but for the others I can't find the way...

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  • $\begingroup$ Hint: $(ba)^n = b(ab)^{n-1}a$ $\endgroup$
    – Randall
    Commented Oct 4, 2020 at 0:22

1 Answer 1

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Let $(bca)^n=e$.

Thus,

$$\begin{align} (abc)^n&=a(bca)^{n-1}bc\\ &=a(bca)^{n-1}(bca)a^{-1}\\ &=a(bca)^na^{-1}\\ &=aea^{-1}\\ &=e. \end{align}$$

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