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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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


Facts used

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


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