# Homomorphisms of $S_3$

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?

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.
• $\mathbb{Z}_2$ is the group with two elements and $S_3/\{e,(1\ \ 2\ \ 3),(1\ \ 3\ \ 2)\}$ is the quotient of the group $S_3$ by the subgroup $\{e,(1\ \ 2\ \ 3),(1\ \ 3\ \ 2)\}$. – José Carlos Santos Jan 21 '19 at 14:38