The Maximum possible order for an element $S_n$ Given the following groups, what is the maximum possible order for an element for 
(a) $S_5$
(b) $S_6$
(c) $S_7$
(d) $S_{10}$
(e) $S_{15}$
My book justifies the answer as


(a) The greatest order is $6$ and comes from a product of disjoint cycles of length 2 and 3
(b) The greatest order is $6$ and comes from a cycle of length $6$


The other answers were justified exactly the same way, that is (c) 12, (d) 30, (e) 105
I do not understand how in (a) we even got the number "6" from $S_5$ and what disjoint cycles they are referring to. Could someone at least justify one for me? 
 A: Consider the permutation $p = (1 2)(3 4 5)$.  It is an element of $S_5$, but it has order 6.  The "disjoint cycles" are $(1 2)$ and $(3 4 5)$, which have lengths of 2 and 3, respectively. 
If you don't understand the "cycle notation" $(1 2)(3 4 5)$ leave a comment and I will explain further.  The short version is that $(1 2)(3 4 5)$ is the permutation which sends $1\mapsto 2$, 
$2\mapsto 1$, 
$3\mapsto 4$, 
$4\mapsto 5$, and 
$5\mapsto 3$.
A: You will have to write out the possible forms a given permutation (expressed as the product of disjoint cycles) can take, and then use the convenient fact that for disjoint cycles $\sigma_{1}, \dots, \sigma_{k} \in S_{n}$, 
$$|\sigma_{1} \dots \sigma_{k}| = \textrm{lcm}(|\sigma_{1}|, \dots, |\sigma_{k}|).$$
For example, in $S_{5}$, you have (up to isomorphism) the following forms that a given permutation (written as the product of disjoint cycles) can take:


*

*$(1 2 3 4 5)$

*$(1 2 3)(4 5)$

*$(1 2 3 4)$

*$(1 2)(34)$

*$(1 2 3)$

*$(1 2)$


Then figure out which of the above forms will have the greatest order.
There is a sequence of values (of Landau's function, $g(n)$) that you can refer to for many values of $n$.
There is a known upper bound on the function:
$$g(n) < e^{n/e}.$$
A0000793: Landau's function g(n): largest order of permutation of n elements, Equivalently, largest lcm of partitions of n. 
