Question: Determine the signature of the elements in $S_4$.
Progress: I've deduced that the permutations in $S_4$ can have the following structure:
4-cycles such as $sgn(1234)=sgn(12)(13)(14)=(-1)^3$ which have odd signature.
Pairs of transpositions such as $sgn(13)(24)=(-1)^2$ which have even signature.
The identity permutation $(1)(2)(3)(4)$. Does this have even signature because we have no transpositions and $0$ is an even number?
3-cycles and a 1-cycles such as $sgn(134)(2)=sgn(13)(14)(2)=(-1)^2?$
Two 1-cycles and a transposition such as $sgn(1)(2)(34)=-1?$
Am I using the definition of signature correctly, specifically for the identity and last two permutations?