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
Best Answer
In general, the order of any $k$-cycle is $k$.
However, if you have a composition of disjoint cycles, say a $k$-cycle with an $l$-cycle, then the order of the composition will be $\mathrm{lcm}(k, l)$.
(Prove this!)