Worksheet 10.4.1 Activity: Working with \(S_5\)
We are close to our goal of finding a degree 5 polynomial whose roots cannot be expressed in a nice way (using radicals and field operations). It turns out that a key step in this goal is understanding the group \(S_5\text{.}\) In this activity, we will get a better feel for this group by examining the “cycle structure” of its elements.
1.
Every element in \(S_5\) can be written as a single cycle or a product of disjoint cycles. Write down all the different possibilities for how these cycles or products of cycles might look (focusing on their “shape” rather than the specific numbers in the cycles).
2.
Does \(S_5\) contain a non-trivial subgroup that contains all the transpositions (2-cycles)? What is it, or why not?
3.
Does \(S_5\) contain a non-trivial subgroup that includes the elements \((12)\text{,}\) \((13)\text{,}\) \((14)\text{,}\) and \((15)\text{?}\) What else would it contain? Hint: what is \((12)(14)(12)\text{?}\)
4.
Does \(S_5\) contain a non-trivial subgroup that contains \((24) \) and \((12345)\text{?}\) Think about what else such a subgroup would contain.
5.
What if you started with a different 2-cycle and a different 5-cycle?Would any pair of 5-cycle and 2-cycle work?
Now let's consider the alternating group \(A_5\text{.}\) Recall this is the group of all permutations in \(S_5\) that can be written as the product of an even number of 2-cycles.
6.
If you write elements of \(A_5\) as the product of disjoint cycles, what sorts of cycle structures do you get?
7.
Does \(A_5\) contain a non-trivial subgroup that contains all the 3-cycles? Hint: show that every pair of transpositions can be written as a product of 3-cycles.
8.
Now consider normal subgroups \(N\) of \(A_5\text{.}\) Remember, a normal subgroup is closed under conjugates (here the conjugate would be \(aba\inv\) where \(a \in A_5\) and \(b \in N\)). Does \(A_5\) contain a non-trivial normal subgroup that contains \((123)\text{?}\)
9.
Look at the different cycle structures of elements in \(A_n\) and start taking conjugates. Will you be able to get \((123)\) starting from any non-identity element?