The Maximum possible order for an element $S_n$ [duplicate]

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?

Solution 1:

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. $(1 2 3 4 5)$
2. $(1 2 3)(4 5)$
3. $(1 2 3 4)$
4. $(1 2)(34)$
5. $(1 2 3)$
6. $(1 2)$

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.

Solution 2:

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 $5\mapsto 3$.