# Number of nth roots is a multiple of n

From Groupprops

This article states a result of the form that one natural number divides another.

View other divisor relations |View congruence conditions

## Contents

## Statement

Suppose is a finite group and is any natural number dividing the order of . Then, the size of the set:

is a multiple of .

Note that since the identity element is itself in this set, the size of the set is at least .

## Related facts

### Stronger facts

- Exactly n elements of order dividing n in a finite solvable group implies the elements form a subgroup
- At most n elements of order dividing n implies every finite subgroup is cyclic
- Number of nth roots of a subgroup is divisible by order of subgroup

### Conjectures

## Facts used

## Proof

The proof follows from fact (1); in fact, the given statement is a special case of fact (1).