Verify that the mapping ({e,(123),(321)} ightarrow 1) and ({(12),(23),(31)} ightarrow-1) preserves the group multiplication operation

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Verify that the mapping \(\{e,(123),(321)\} \rightarrow 1\) and \(\{(12),(23),(31)\} \rightarrow-1\) preserves the group multiplication operation for the permutation group \(\mathrm{S}_{3}\).

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