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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?

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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.

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  • $\begingroup$ I find it hard to understand the notation Z2(≃S3/{e,(1 2 3),(1 3 2)}), can you elaborate, please? $\endgroup$ – Alternatic Jan 21 '19 at 14:37
  • $\begingroup$ $\mathbb{Z}_2$ 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 $\{e,(1\ \ 2\ \ 3),(1\ \ 3\ \ 2)\}$. $\endgroup$ – José Carlos Santos Jan 21 '19 at 14:38

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