Highest order element in $S_5$

Question

What is the largest order of an element in the group of permutations of 5 objects ?

Any idea how to proceed ?

Answer 1

First of all: note that many elements have the same order (all 2-cycles, all 3-cycles, a 2-cycle and 3-cycle combined, etc...). You only need to check every "form" of element that is available. I recommend using the disjoint cycle notation for this.

The element you seek might be composed of multiple cycles, and a single $n$-cycle has order $n$. Out of those, the highest order is a $5$-cycle with order $5$.

So what is the highest order we can create by combining multiple cycles? The order of such combination will be the $\text{lcm}$ of the order of the subcycles. Also note that $1$-cycles don't do anything, so we only need to check combinations of cycles of length $\geq 2$.

1. $2$-cycle + $2$-cycle: order $\text{lcm}(2,2) = 2$.
2. $2$-cycle + $3$-cycle: order $\text{lcm}(2,3) = 6$.

Hence the answer is: any combination of a $2$-cycle and $3$-cycle, e.g. $(1 2)(3 4 5)$.