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

Question

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

Hint: $(ba)^n = b(ab)^{n-1}a$ – Randall Oct 04 '20 at 00:22 — Randall, Oct 04 '20 at 00:22

score 1 · Answer 1 · edited Oct 04 '20 at 00:57

1

Let $(bca)^n=e$.

Thus,

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

edited Oct 04 '20 at 00:57

Shaun

44,997

answered Oct 04 '20 at 00:44

Michael Rozenberg

194,933

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

1 Answers1