## Lagrange's Theorem

If is a finite subgroup with n elements, the number of elements in any subgroup of must divide This is Lagrange's Theorem. Formally, Lagrange's Theorem states,

The order of any subgroup of a group must divide the order of If has order 10, the only possible orders of any subgroup of are 1, 2, 5 or 10. 1 is the order of the trivial group consisting of the identity element, and 10 is the order of G, which may be considered a subgroup of itself.

Lagrange's theorem has several corollaries.

Corollary 1 The order of an element is the least value of satisfying The order of divides the order of This can be seen by considering the set generating by repeatdly composing a with itself to form the set This is a cyclic subgroup of with elements and divides the order of but so the order of divides the order of Corollary 2 A group with prime order has only two subgroups. The only divisors of a prime are 1, corresponding to the trivial subgroup, and corresponding to the group Corollary 3 Every group of prime order is cyclic. The order of every element divides the order of the group, but a group of prime order the only divisors of the order of the group are 1 and Every element of the group except (which has order 1) then has order The converse of Lagranges theorem is not true. It is not the fact that a group has a subgroup for every divisor of the order of the group. the subgroup of consisting of the even permutations, has order 12 but no subgroup of order 6.