The famous Lagrange’s Theorem states that, in any finite group , the order of any element divides the order of . A natural follow up to this theorem is whether or not the converse holds. That is, if divides the order of , does there exist an element of order ? Unfortunately the converse is, in general, false. Examples are easy to come by; take, for instance, and look for an element of order , .
While the converse to Lagrange’s Theorem is false in general, Cauchy’s Theorem tells us that there are special circumstances in which the converse is true. Namely, if is a finite group and is a prime number dividing , then we can guarantee that some element in has order . This means, for example, that in the symmetric group we can find an element of order for all (it is easy to come up with the elements here, just take a -cycle. In general Cauchy’s Theorem does not tell you how to find such elements).
There are several proofs of Cauchy’s Theorem available. The standard first proof uses induction on the size of , and while it is instructive, I find it to be a bit messy. Below I reproduce a (in my opinion) more elegant proof using group actions. The original version of this proof is due to James McKay.
Proof of Cauchy’s Theorem: Let be a finite group, a prime with dividing . Consider the set
In other words, consists of tuples from whose ordered product equals the identity. Note that once we choose , the element is forced to equal , and thus .
Now we make the following observation: if , then any cyclic permutation of the product must also yield the identity (to see this, use the relation that if then and induct). Thus admits an action by the cyclic group . If , then by the Orbit-Stabilizer Theorem the orbit of must have size or . What does it mean for to have an orbit of size ? If this is the case, then it must be that with ; in otherwords, represents an element with . As is prime, if we can find such a which is not the identity element, we will have found our element of order .
To finish, we use the fact that the orbits of a group action partition the set on which the group is acting. Thus we have
We know that is contained in since , and thus there is one orbit of size . Thus
Now as mentioned previously. Further, is divisible by , and thus the right hand side above is divisible by . It follows immediately that the number of orbits of size is a multiple of , and as we have already found one, there must be at least . Thus we can guarantee an element of order as desired.
As always, thanks for reading!