Theorem

Cauchy's Theorem (Group Theory)

Algebra

Cauchy's Theorem states that if a prime number p divides the order of a finite group, then that group contains an element of order p. Proved by Augustin-Louis Cauchy, it is a basic structural result in group theory and a special case of the later, more general Sylow Theorems.

Facts

Statement

If G is a finite group and p is a prime number dividing the order of G, then G contains at least one element of order p. The result is a partial converse to Lagrange's theorem and was proved by Augustin-Louis Cauchy in 1845. 1

Proof Year

1845 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Inequality 1

Connections

Sources

1. Cauchy's Theorem (Group Theory) (Wikipedia)

Wikimedia Foundation

lead section, first paragraph
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 xp = e, where e is the identity element of G. It is named after Augustin-Louis Cauchy, who discovered it in 1845.
In Group: Group Theory, Article text
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.

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Cauchy's Theorem (Group Theory)

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