Homomorphisms of $S_3$

Question

What are all the groups (not counting the isomorphic ones) that can be a homomorphic image of $S_3$?

So here's what I have come up with so far:

• First of all, $S_3$ has 3! = 6 elements, which are $\{ id, (12), (23), (13), (123), (132) \}$

  • The order of identity is one, the order of 2-cycles is 2 and the order of 3-cycles is 3.
• The order of the group that is an image of $S_3$ has to be of order 6, 3, 2 or 1
  • If it's of order 1, than all elements are projected onto the same element, that is trivial
  • Can it also be of order 4? Since GCD(6, 4) > 1

But where do I go from here? How can I be certain that I've found all the possible groups?

Answer 1

The group $S_3$ only has two normal subgroups distinct from $\{e\}$: $\{e,(1\ \ 2\ \ 3),(1\ \ 3\ \ 2)\}$, and $S_3$ itself. Therefore, the groups that you're after are $\mathbb{Z}_2(\simeq S_3/\{e,(1\ \ 2\ \ 3),(1\ \ 3\ \ 2)\}$) and the trivial group.