In Artin's Algebra under the section on the Class Equation is the exercise
The class equation of a group $G$ is $1+4+5+5+5$. (a) Does $G$ have a subgroup of order $5$? If so, is it normal? (b) Does $G$ have a subgroup of order $4$? If so, is it normal?
I can solve this exercise. Since the group is obviously of order $20=2^25$, then by the Sylow theorems, it has a Sylow $2$-subgroup of order $4$ and similarly a Sylow $5$-subgroup of order $5$. The Sylow $5$-subgroup is obviously normal, but the Sylow $2$-subgroup is not (see this question). So the answer to (a) is yes and it is normal, and the answer to (b) is yes but it is not normal.
But the book does not introduce the Sylow theorems until a bit later on, and as mentioned the exercise is listed under the section on Class Equations. So presumably, there is way to do this exercise without using the Sylow theorems. Can someone give me a hint? Thanks in advance!
Let $G$ act on itself through conjugation. Then the orbits are exactly the different classes of the group $G$. Notice that that for each element of the orbit of four elements there is a stabilizer of order 5 which forms a subgroup. It is easy to show that this subgroup is fixed under conjugation, since the elements in the class of order 4 and the identity all belong to the stabilizer of the elements of class of order 4.
However this is not for the stabilizers of the elements in classes of $5$ elements since they must include the identity and some other classes to be normal which can never add up to 4.