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Lagrange's theorem (group theory)

    ::''This article is about Lagrange's theorem in group theory See also Lagrange's theorem (number theory)''
    Lagrange's theorem in the mathematics of group theory 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 [1] (the number of left cosets of H in G) If we write this statement as
    |G| = [2] · |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 (ie 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 cyclic 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
    The converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of |G| there does not necessarily exist a subgroup of G with order d The smallest example is the alternating group G = A4 which has 12 elements but no subgroup of order 6. However if G is abelian then there always exists a subgroup of order d

    See also

    • Sylow's theorem