Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order (number of elements) of every subgroup H of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. 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. To show any two left cosets have the same cardinality, it suffices to demonstrate a bijection between them. Suppose aH and bH are two left cosets of H. Then define a map f : aH → bH by setting f(x) = ba−1x. This map is bijective because it has an inverse given by f − 1 ( y ) = a b − 1 y . {displaystyle f^{-1}(y)=ab^{-1}y{mbox{.}}} 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 allow G and H to be infinite, and write this statement as then, seen as a statement about cardinal numbers, it is equivalent to the axiom of choice. A consequence of the theorem is that the order of any element a of a finite group (i.e. the smallest positive integer number k with ak = e, where e is the identity element of the group) 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 This can be used to prove Fermat's little theorem and its generalization, Euler's theorem. These special cases were known long before the general theorem was proved. The theorem also shows that any group of prime order is cyclic and simple. This in turn can be used to prove Wilson's theorem, that if p is prime then p is a factor of ( p − 1 ) ! + 1 {displaystyle (p-1)!+1} . Lagrange's theorem can also be used to show that there are infinitely many primes: if there were a largest prime p, then a prime divisor q of the Mersenne number 2 p − 1 {displaystyle 2^{p}-1} would be such that the order of 2 in the multiplicative group ( Z / q Z ) ∗ {displaystyle (mathbb {Z} /qmathbb {Z} )^{*}} (see modular arithmetic) divides the order of ( Z / q Z ) ∗ {displaystyle (mathbb {Z} /qmathbb {Z} )^{*}} , which is q − 1 {displaystyle q-1} . Hence p < q {displaystyle p<q} , contradicting the assumption that p is the largest prime.