As homework the teacher asks us to determine how many elements are there in $\langle (123) , (234) \rangle \subset S_4$ .
I've started doing all the multiplications between the elements, and I've counted so far $9$ different elements. But I think there is a easier way to determine the order of this subgroup of $S_4$. My argument is:
$(123)$, $(234)$ are even permutations, so they can generate only even permutations. So our subgroup has at most $12$ elements. I've counted so far $9$ distinct elements, so using the Lagrange Theorem it must have 12 elements.
Is it correct this argument?