Question

Write a hypothesis for the order of an arbitrary element of Sn

4. Write a hypothesis for the order of express any permutation as a arbitrary element of Sn. (Hint: we know that we can an product of disjoint cycles).

Answer 1

First note that, any element $\sigma\in S_n$ can be expressed as a product of disjoint cycles. This factorization is unique up to reordering of cycles.

Now order of any element in can be calculated from the following Lemma.

ST

Lemma: Proof. Let  
= T1T2...TE ST n
  be the decomposition of $\sigma$ into disjoint cycles of lengths
2: 1:
, respectively. Then the order of $\sigma$ is the least common multiple of  
2: 1:
, that means,

Order(o lem(l1, l2, ...lt)

Proof: First notice that, for any integer we get

$\sigma^m=\tau_{1}^m \tau_{2}^m \ldots \tau_{k}^m$

This gives us

id T Tid T id, Vi 1,2,., m m m

Now let  
Order(op
. Then we get $\sigma^p=\operatorname{id}$ . Thus we have

$\tau_i^p=\operatorname{id}, \ \forall i=1,2,\ldots,m$

Also we have $\operatorname{Order}(\tau_i)=\ell_i$ , for
.., m 2=1,2,
. This gives us $\ell_i$ divides , for
.., m 2=1,2,
.

That means, $\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)$ divides .

Also let, then

Iem(, l2,l4) = tl4l2*** ,

$\tau_{1}^{\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)} \tau_{2}^{\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)} \ldots \tau_{k}^{\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)}=\operatorname{id}\implies \sigma^{\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)}=\operatorname{id}$

Thus we get p divides $\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)$ .

Hence $p=\operatorname{lcm}( \ell_1,\ell_2,\ldots, \ell_k)$