# Thread: Order of an element

1. ## Order 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 S5xS7?

2. ## Re: Order of an element

As my calculations, I think the maximum order of an element of $S_{5} \times S_{7}$ 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)).

3. ## Re: 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.

4. ## Re: 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 Sn, the order of any permutation x which is a product of k disjoint cycles of finite lengths m1 ,...,mk is the least common multiple of these lengths , i.e. | x | = lcm( m1 ,...,mk ) and we have n=m1+...+mk(by considering cycles of length 1).

5. ## Re: Order of an element

It makes sense, thank you very much.