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}?

Printable View

- April 18th 2013, 03:44 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}? - April 18th 2013, 08:54 AMxixiRe: Order of an element
As my calculations, I think the maximum order of an element of 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)).

- April 18th 2013, 09:18 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. - April 18th 2013, 10:55 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). - April 18th 2013, 07:00 PMchristianwosRe: Order of an element
It makes sense, thank you very much.