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The Question: Let $G_t = \{x \in G\ |\ o(x) = t\}$. Write down $G_t$ for $G = S_8$ and all $t$ and determine the highest order for an element in $S_8$.

So for starters, $S_8$ is the set of all bijective functions from $\{1,2,3,4,5,6,7,8\}$ to itself and $o(x)$ is the order of the element $x$.

Attempt at a solution

My initial thought is that the highest order for an element in $S_8$ is 8 since for the function $$f(n) = \begin{cases} n+1, & \ 1\le n \le 7 \\ 1, & \ n=8 \end{cases}$$

the order would be 8.

I am totally lost on the first part of the question where it asks us to find $G_t$ for all $t$. If this is asking what I think it's asking, I'll have to write out tons of functions. Is there a way to write this quickly and succinctly?

Also, here is my professor's solution, which is essentially Greek to me. If anyone could explain what she's trying to say here, that would be fantastic:

Practice Exam Solution #1d

Thanks in advance!

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    Are you familiar with the representation of permutations as products of disjoint cycles? Judging from your teacher's solution that should have been covered. Mind you, the teacher's list about $G_6$ is incomplete. It is missing $(a_1,a_2,a_3)(a_4,a_5)$,$(a_1,a_2,a_3)(a_4,a_5,a_6)(a_7,a_8)$ and $(a_1,a_2,a_3)(a_4,a_5)(a_6,a_7)$. – 2017-02-13
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    But, yeah. You are right in the sense that there are a total of $40320$ permutations. The idea is to compress that list to a handful of entries based on the cycle type of the permutation. – 2017-02-13
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    No, that terminology does not sound familiar. I understand what each of those words mean individually, but I don't understand what $(a_1,a_2,a_3)(a_4,a_5)$ means, let alone how this might answer the question. Can you provide some insight? – 2017-02-13
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    The element $(1\;2\;3\;4\;5)(6\;7\;8)$ has order 15, which you can painfully check so the highest order element in $S_8$ certainly cannot be 8. Think about disjoint cycles - meaning cycles which contain no common element. Do they commute? What does this mean in terms of their order? What does the length of a cycle have to do with its order? Can you then see how to find an element of largest possible order? – 2017-02-13

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Hint

Consider all possible decompositions of $\sigma\in S_8$ as a product of cycles with pairwise disjoint supports and compute order in each case (recall that the order of any permutation is given by the lowest common multiple of the orders of the cycles involved in the decomposition) :

$(ab)$ order 2

$(ab)(cd)$ order 2

$(ab)(cd)(ef)$ order 2

$(ab)(cd)(ef)(gh)$ order 2

$(abc)$ order 3

$(abc)(de)$ order 6

$(abc)(de)(fg)$ order 6

$(abc)(def)$ order 3

$(abc)(def)(gh)$ order 6

$(abcd)$

$(abcd)(ef)$

$(abcd)(ef)(gh)$

$(abcd)(efg)$

$(abcd)(efgh)$

and so on ... (still looks heavy, though !...)

Remark

Your initial thought (that the order of any permutation should be less than 8) is wrong : consider for example $(123)(45678)$, whose order is $15$.

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    I'm not entirely sure about the notation here. As I mentioned in a comment above, this concept of disjoint cycles isn't quite clear to me. But I'll take a guess--let me know if I'm on the right track: (abc)(defgh) means that a function would map $a \to b, b \to c, c \to a$ and $d \to e, e \to f, f \to g, g \to h, h \to d$. Is this correct? – 2017-02-13
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    Correct, @BlakeSplitter. That's what the cycle notation means. When you have an $n$-cycle $(a_1a_2\ldots a_n)$ it is of order $n$. When you have a product of two or more **disjoint** cycles, the order of the permutation is the l.c.m. of the lengths of the cycles. This is because disjoint cycles commute, so for example $(12345)(678)$ applied $k$ times is the composition of $(12345)$ applied $k$ times and $(678)$ applied $k$ times. That rule does not apply, when the cycles overlap. – 2017-02-13
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    Okay, I think I'm starting to understand the disjoint cycle theory. How then could I ensure that 15 was the largest order? Moreover, what are the rules governing cycles that $do$ overlap? – 2017-02-13
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    @BlakeSplitter: The underlying theorem asserts that any permutation can be written as the (commutative) product of cycle whose supports are pairwise disjoints (which means : *non* overlapping cycles). – 2017-02-13