Language of all sequences of permutations whose product is the identity Let $\Sigma$ be the set of all permutations in $S_n$. What is the minimum number of states in a DFA accepting the language of all words over $\Sigma$ which multiply to the identity permutation?

For example, if $n=2$, then $\Sigma$ consists of two mappings, the identity mapping $\iota$ and the transposition $\tau$. The language in this case consists of all words containing an even number of $\tau$'s.
You haven't stated what happens to the empty word – I'm assuming it's accepted.

It is easy to check that the equivalence classes of the Myhill–Nerode relation correspond to all words multiplying to a certain permutation. Therefore the minimal DFA contains $n!$ states.

If the empty word is not allowed, there are $n!+1$ equivalence classes, one of them consisting only of $\epsilon$.