Cauchy's theorem (group theory)

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In mathematics, specifically group theory, Cauchy's theorem 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 such that p is the smallest positive integer with x^p = e, where e is the identity element of G. It is named after Augustin-Louis Cauchy, who discovered it in 1845.^[1][2]

The theorem is a partial converse to Lagrange's theorem, which states that the order of any subgroup of a finite group G divides the order of G. In general, not every divisor of ${\displaystyle |G|}$ arises as the order of a subgroup of ${\displaystyle G}$.^[3] Cauchy's theorem states 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.

Cauchy's theorem is generalized by Sylow's first theorem, which implies that if pⁿ is the maximal power of p dividing the order of G, then G has a subgroup of order pⁿ (and using the fact that a p-group is solvable, one can show that G has subgroups of order p^r for any r less than or equal to n).