Prove that set of permutation on $3$ elements is not isomorphic to $(\Bbb Z_6,+)$.

Question

Prove that set of permutation on $3$ elements is not isomorphic to $(\Bbb Z_6,+ )$ (the group with $+$ on $\Bbb Z_6$).

Hello everyone,

I tried to build a function and show that associativity doesnt work because of the permutation but I don't know how to write the proof, I tried using table or examples.

Can someone help please ?

Thank you

Answer 1

The group $S_3$ of permutations of $3$ elements is not abelian.

For instance, consider the transpositions $\tau=(1\,2)$, $\tau'=(2\,3)$. Then \begin{align} &&\tau\tau'&=(1\,2\,3)&\qquad& (\text{denote it }\gamma)\\ &\text{whereas }\qquad\qquad &\tau'\tau&=(1\,3\,2)=\gamma^{-1} \end{align}

Answer 2

Hint Show that $S_3$ is not cyclic.

Answer 3

Note that $(\Bbb Z_6, +)$ has an element of order six, whereas $S_3$ does not. Isomorphisms preserve orders of elements.