Worksheet 11.1.2 Activity: Orders of Permutations
The order of an element \(g\) in a group \(G\) is the least natural number \(n\) such that \(g^n = e\text{,}\) if such a number exists (otherwise we say the order of \(g\) is infinite).
1.
Find the orders of the elements of \(S_5\) below:
The orders are 2, 3, 4, and 5, respectively.
2.
Find an element of \(S_5\) that has an order different from those found above.
We will need an element that is not just a cycle, since the longest cycle is of length 5. So how about \((12)(345)\text{.}\) This will have order 6.
3.
Let \(\alpha\) be an element of \(S_5\text{.}\) What is \(\alpha^{120}\text{?}\)
No matter what element we pick, \(\alpha^{120}\) will always be \((1)\text{.}\) This will work in general, but you can also see this by considering the orders of elements, which must all divide the size of the group.
4.
Is there an element in \(S_5\) that has order \(120\text{?}\)
No there is not. If there were, this would say that \(S_5\) was cyclic, and would thus be abelian, which we know is not the case. Alternatively, just look at how to write elements as disjoint cycles.
5.
What is the largest order of any element in \(S_5\text{?}\)
The largest order will be 6, for elements that are the disjoint product of a 2-cycle and 3-cycle.