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.
View other divisor relations |View congruence conditions


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.

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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).