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Lagrange's theorem

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In mathematics, Lagrange's theorem 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 in G. 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 : aHbH 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] (the number of left cosets of H in G). 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 a of a finite group (i.e. the smallest positive integer k with ak = e) divides the order of that group, since the order of a is equal to the order of the subgroup generated by a. If the group has n elements, it follows

an = e.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.



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