# Frobenius conjecture on nth roots

From Groupprops

This article is about a conjecture in the following area in/related to group theory: finite groups. View all conjectures and open problems

## Statement

Suppose is a finite group and is a natural number dividing the order of . Suppose the number of roots in , i.e., the number of elements such that , is exactly .

The **Frobenius conjecture on nth roots** states that in that case, that set of roots must be a subgroup of .

## Related facts

- Exactly n elements of order dividing n in a finite solvable group implies the elements form a subgroup: This is precisely the Frobenius conjecture in the case of finite solvable groups.
- Number of nth roots is a multiple of n
- Number of nth roots of any conjugacy class is a multiple of n
- At most n elements of order dividing n implies every finite subgroup is cyclic