Theorem of Lagrange

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In mathematics, the theorem of Lagrange states that if G is a finite group and H is a subgroup of G, then the order (that is, the number of elements) of H divides the order of G.

This can be shown using the concept of left cosets of H (see under subgroup). The left cosets are the equivalence classes of a certain equivalence relation on G and therefore form a partition of G. If we can show that all cosets of H have the same number of elements, then we are done, since H itself is a coset of H. 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 f -1(y) = ab-1y.

This proof also shows that the quotient of the orders |G| / |H| is equal to the index [G:H]. If we write this statement as |G| = [G:H] · |H|, then, interpreted as a statement about cardinal numbers, it remains true even for infinite groups G and H.

A consequence of the theorem is that the order of any element of a finite group divides the order of that group, and in particular

an = e

for any element a of any group with n elements. This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.