= Solution
Compose <permutations> from right to left. Tracking the four symbols through the two <permutation cycles> gives $1\mapsto2\mapsto1$ and $3\mapsto4\mapsto3$. Thus the <cycle decomposition of a permutation> is
$$
\boxed{(123)(234)=(12)(34).}
$$
The disjoint transpositions commute, and squaring their product gives the <identity element>, while the product itself is nonidentity. Its <order of a group element> is therefore \b[two]. Each transposition has negative <sign of a permutation>, so the product has sign $(-1)^2=\boxed{+1}$: it is an <even permutation>.
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