I am a bit confused about this. Could someone guide me in the right direction? Thank you.
What is the maximum order of an element of S_{5}xS_{7}?
As my calculations, I think the maximum order of an element of $\displaystyle S_{5} \times S_{7}$ is 60, which is the order of an element with a cycle of length 5 in the first component and a permutation which is product of a cycle of length 3 and a cycle of length 4, in the second component. One of such element is ((12345),(123)(4567)).
The order of each element (g, h) in G × H is the least common multiple of the orders of g and h:
| (g, h) | = lcm( | g |, | h | ).
Besides, in S_{n}, the order of any permutation x which is a product of k disjoint cycles of finite lengths m_{1} ,...,m_{k} is the least common multiple of these lengths , i.e. | x | = lcm( m_{1} ,...,m_{k} ) and we have n=m_{1}+...+m_{k}(by considering cycles of length 1).