Lagrange's theorem (group theory)

A foundational result in finite group theory: the order of any subgroup divides the order of the group, and the quotient gives the number of cosets. Consequences include constraints on element orders and prime-order groups.

Lagrange's theorem is a basic structural result in finite group theory that relates the sizes of a group and its subgroups. Informally, if G is a finite group and H is a subgroup, then the number of elements in H (the order of H) divides the number of elements in G (the order of G). The quotient |G|/|H| is the index of H in G and equals the number of distinct left (or right) cosets of H.

Statement and terminology

Let G be a finite group and H a subgroup of G. Lagrange's theorem asserts that |H| divides |G| and that |G| = [G:H]·|H|, where [G:H] denotes the index of H. A coset is a translate of H by a group element: a left coset has the form gH and a right coset Hg. All cosets of H have the same cardinality, so they partition G into equally sized blocks. For further reading on the formal statement see Lagrange's theorem and for the notion of a finite group consult finite group.

Sketch of the proof

The proof uses the partition of G into cosets of H. Fix H and consider the set of left cosets {gH : g in G}. Distinct cosets are disjoint and every element of G lies in some coset, so these cosets form a partition of G. Each coset has exactly |H| elements because the map h -> gh is a bijection from H to gH. Counting elements of G by summing the sizes of the cosets gives |G| = [G:H]·|H|, and hence |H| divides |G|.

$g^{k}=e$

Consequences and examples

Order of an element: if g is an element of G, the order of g (the size of the cyclic subgroup generated by g) divides |G|. In particular g^|G| = e when G is finite.
Prime-order groups: every group whose order is a prime p is cyclic and has no nontrivial proper subgroups; such groups are isomorphic to the cyclic group of order p. This follows because any non-identity element must generate the whole group.
Index and normality: if a subgroup has index 2, it must be normal in G because left and right cosets coincide in that case.
Orbit-stabilizer and related counting arguments use the same partitioning idea and generalize the counting principle behind Lagrange's theorem.

Limitations and notable facts

Lagrange's theorem gives a necessary condition for a subgroup order, but not a sufficient one: a divisor of |G| need not correspond to an actual subgroup. A standard example is the alternating group A4 (order 12), which has no subgroup of order 6 even though 6 divides 12. Other results, such as Cauchy's theorem and Sylow theorems, provide additional guarantees about existence of subgroups of certain prime-power orders, but these require separate hypotheses.

Finally, Lagrange's theorem applies only to finite groups; for infinite groups the relationship between sizes is measured by cardinalities and different phenomena occur. For more on cosets and related constructions see cosets, and for historical background consult Joseph-Louis Lagrange.