How to find the order of the elements of $A_4$? Just wondering how to find the order of each element in this group:
$A_4 = \{e,(123),(132),(124),(142),(134),(143),(234),(243),(12)(34),(13)(24),(14)(23)\}$
I tried writing each elements not in disjoint cycle but it didn't look right to me. I got 3 for all the cycles with 3, and 4 for the last cycles

The order of any $k$-cycle is $k$, try seeing why by trying a few examples for yourself. Now the disjoint product of $k$-cycles commutes; therefore, if we have the product of $n$, $k$-cycles, $x_1x_2,x_3,\cdots$, we would have
$$
(x_1x_2\cdots x_n)^k=x_1^kx_2^k\cdots x_n^k
$$
so that the order of a product of disjoint $k$-cycles (not necessarily all of the same length) must be the least common multiple of their orders (their length). Since we write the elements of $A_n$ in disjoint cycle notation, this makes the computation of the orders a quick task. Just be sure to understand why the above things are true (try proving them formally for yourself!).
