Cauchy's theorem (group theory)

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Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin Louis Cauchy. It states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G so that p is the lowest non-zero number with x^p = e, where e is the identity element.

The theorem is a partial converse of Lagrange's theorem, which states that the order of any subgroup of a finite group G divides the order of G. Cauchy's theorem implies that for any prime divisor p of the order of G, there is a subgroup of G whose order is p - the cyclic group generated by the element in Cauchy's theorem.