# Surjective Group Homomorphism with the Symmetric & Integers

(A) Find all of the subgroups of $S_3$ (permutations on 3 letters) which have order 2.

$\textbf{ANSWER:}$ ( 1 2 ), ( 2 3 ), and ( 1 3 )

(B) Which of the subgroups in (A) are normal?

$\textbf{ANSWER:}$ None because the group doesn't have the same order and the same number of normal subgroups.

(C) Is there a surjective group homomorphism from $S_3$ to $\textbf{Z}_3$ (the integers modulo 3)?

I provide the previous part of the problem (A) and (B), to see if it helps solve (C)?

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Thanks for including the first two parts. They'll help. You should clarify your argument for (B). Once we've got (B) down, think about the isomorphism theorem $G/\ker\varphi\cong\operatorname{im}\varphi$. So should there be surjective $\varphi:S^3\to \Bbb{Z}_3$, we'd have to have $\Bbb{Z}_3$ as a quotient of $S^3$. Now what does the fact that you've shown there are no order-2 normal subgroups of $S^3$ tell you about the existence of such a quotient?