Number of nth roots is a multiple of n

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

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

\{g \in G \mid g^n = e \}

is a multiple of n.

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

Related facts

Stronger facts

Other related facts

Conjectures

Facts used

  1. Number of nth roots of any conjugacy class is a multiple of n

Proof

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