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}?

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- Apr 18th 2013, 02:44 AMchristianwosOrder of an element
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}? - Apr 18th 2013, 07:54 AMxixiRe: Order of an element
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)).

- Apr 18th 2013, 08:18 AMchristianwosRe: Order of an element
Thank you. I am still confused. Why shouldn't be an element of order 35 = 5x7 then? Why did you split the seven into three and four?

thanks again for your help. - Apr 18th 2013, 09:55 AMxixiRe: Order of an element
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). - Apr 18th 2013, 06:00 PMchristianwosRe: Order of an element
It makes sense, thank you very much.