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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.

Nicky Hekster
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orangefruits
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  • To be clear, are you asking for a group who only has $a,b,c$? Or are you asking for a group who has elements $a,b,c$ among possibly many others? These two questions give different results. All groups of order three are abelian. There exist nonabelian groups of order greater than 3 (the smallest example being of order 4). See examples. – JMoravitz Apr 17 '23 at 12:19
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    Choose b and c so that they don’t commute and then take a the inverse of bc. – Rafi Apr 17 '23 at 12:20
  • Do you further want the group to be the group of symmetries of a shape? – ronno Apr 17 '23 at 13:00

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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 $3$.

Dietrich Burde
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