Lagrange's Theorem (group Theory) - Proof of Lagrange's Theorem

Proof of Lagrange's Theorem

This can be shown using the concept of left cosets of H in G. The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. Specifically, x and y in G are related if and only if there exists h in H such that x = yh. If we can show that all cosets of H have the same number of elements, then each coset of H has precisely |H| elements. We are then done since the order of H times the number of cosets is equal to the number of elements in G, thereby proving that the order of H divides the order of G. Now, if aH and bH are two left cosets of H, we can define a map f : aH → bH by setting f(x) = ba−1x. This map is bijective because its inverse is given by

This proof also shows that the quotient of the orders |G| / |H| is equal to the index (the number of left cosets of H in G). If we write this statement as

then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice.