= Normal subgroups and quotients of the symmetric group on three letters
The <conjugacy classes of the symmetric group on three letters> have sizes $1,3,2$: the identity, the three transpositions and the two 3-cycles. A <normal subgroup> is a union of whole <conjugacy classes> and contains the identity. Its candidate orders are therefore $1,4,3,6$. <Lagrange's theorem> excludes four, leaving precisely $\{e\}$, $A_3$ and $S_3$. The order-three subgroup is the cyclic group of the two 3-cycles and the identity; it is also the <kernel of a group homomorphism> of the sign map.
The <first isomorphism theorem> gives the corresponding quotients $S_3$, $C_2$ and the trivial group. Thus these are exactly the possible images of surjective <group homomorphisms> from $S_3$. Each occurs, by the identity map, the sign map and the constant identity map, respectively.
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