Ifis a finite subgroup with n elements, the number of elements in any subgroup ofmust divideThis is Lagrange's Theorem. Formally, Lagrange's Theorem states,
The order of any subgroup of a groupmust divide the order of
Ifhas order 10, the only possible orders of any subgroup ofare 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 elementis the least value ofsatisfyingThe order ofdivides the order ofThis can be seen by considering the set generating by repeatdly composing a with itself to form the setThis is a cyclic subgroup of withelements anddivides the order ofbutso the order ofdivides the order of
Corollary 2 A group with prime order has only two subgroups. The only divisors of a primeare 1, corresponding to the trivial subgroup, andcorresponding 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 orderthe 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 ofconsisting of the even permutations, has order 12 but no subgroup of order 6.