Could you provide me with an example of a group which has elements $a, b$ and $c$ such that the order of $abc$ is different from the order of $acb$?

Question

I have tried messing with different symmetries and permutations but I am definitely stuck. What is the catch? Do I need to consider shapes that are more than 2 dimensional? Do I need to consider interactions between multilpe objects?

How does one arrive at such group, and what does it represent?

Thank you very much.

Answer 1

You were already on the right track with permutations, but didn't try enough.

Take the symmetric group $S_3$ and let $a=(23)$, $b=(12)$ and $c=(123)$. Then $$ abc=(23)(12)(123)=(132)(123)=(1), $$ but $$ acb=(23)(123)(12)=(13)(12)=(123). $$ So $abc$ has order $1$, whereas $acb$ has order $3$.