Cauchy theorem (group theory)

Cauchy's Theorem in group theory states that if a finite group has a prime number \( p \) as a divisor of its number of elements (called the order of the group), then there is at least one subgroup within that group that contains \( p \) elements. This means that whenever you have a group made up of certain elements, if its total number of elements can be divided by a prime number, you can always find a smaller group formed by that prime number of elements. It highlights the structural richness of groups in mathematics.