There are only two groups of order six, up to isomorphism: $\mathbb Z_6$ and $S_3$.

Question

Let $G$ be group with order $6$. Prove that either $G$ and $\Bbb Z_{6}$ are isomorphic binary structure or $G$ and $S_{3}$ are isomorphic binary structure.

I know that for isomorphic binary structure, we define a function between groups and we should check homomorphism property and bijection. But I can not define a function between them. Please help me, if you have any good idea.

Answer 1

Hint: Either your group has an element of order $6$ or not. If it has, then it is isomorphic to $\mathbb{Z}_6$. Otherwise, which are the possible orders of elemets of $G$?