 Solution (source code)

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= Solution

\b[True.] The <sign of a permutation> is a <group homomorphism> $\operatorname{sgn}:S_3\to\{1,-1\}$, as follows, for example, from the multiplicative <determinant> of <permutation matrices>. Its <kernel of a group homomorphism> is the <alternating group>
$$
\boxed{A_3=\{e,(123),(132)\},\qquad |A_3|=3.}
$$
A transposition maps to $-1$, so this homomorphism is onto.

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