Finding order of permutations

Question

Consider the following disjoint cycles: (1 3 2)(6 4) (7 5 9 8) I'm aware that to find the order of a permutation we take the lcm. However, here we have three disjoint cycles. So do we first take the lcm of the first two: lcm (3, 2) = 6, and then the lcm of (6, 4) = 24? Thanks

Answer 1

You've got the right idea, but note that $\text{lcm }(4, 6) = 12$. And the least common multiple of $(6, 12)$ is $12$.

The least common multiple operator is associative, and so it can take three arguments: $\operatorname {lcm} (a, b, c) =\operatorname{lcm}(\operatorname{lcm}(a,b),c) = \operatorname{lcm}(a, \operatorname{lcm}(b,c))$

Answer 2

We can define the $lcm$ of any set of positive integers in the same way we define it for a pair of them. Namely, $lcm(a_1,a_2,\dots,a_n)=k$, where each $a_i$ divides $k$, and furthermore if $j$ is another integer which is divisible by each $a_i$, then we require that $k$ also divides $j$.

It turns out that, as you have done, computing this $lcm$ is equivalent to doing it inductively, two numbers at a time.

So for example, as has been shown your permutation has order $12$; we may also compute the order of $(1,3)(2,4,5)(6,8,7)(10,9,11,12)(13,14,15,16,17)\;$ as the $lcm$ of $2,3,3,4$, and $5$. By your method, we can write $$ lcm(2,3,3,4,5)=lcm(lcm(2,3),3,4,5)=lcm(6,3,4,5)=\dots=60 $$