Lagrange's theorem (group theory)

In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a divisor of | G | {\displaystyle |G|} . That is, the order (number of elements) of every subgroup divides the order of the whole group. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup H {\displaystyle H} of a finite group G {\displaystyle G} , not only is | G | / | H | {\displaystyle |G|/|H|} an integer, but its value is the index [ G : H ] {\displaystyle [G:H]} , defined as the number of left cosets of H {\displaystyle H} in G {\displaystyle G} .

This variant holds even if G {\displaystyle G} is infinite, provided that | G | {\displaystyle |G|} , | H | {\displaystyle |H|} , and [ G : H ] {\displaystyle [G:H]} are interpreted as cardinal numbers.

Proof The left cosets of H in G are the equivalence classes of a certain equivalence relation on G: specifically, call x and y in G equivalent if there exists h in H such that x = yh. Therefore, the set of left cosets forms a partition of G. Each left coset aH has the same cardinality as H because x ↦ a x {\displaystyle x\mapsto ax} defines a bijection H → a H {\displaystyle H\to aH} (the inverse is y ↦ a − 1 y {\displaystyle y\mapsto a^{-1}y} ). The number of left cosets is the index [G : H]. By the previous three sentences,

| G | = [ G : H ] ⋅ | H | . {\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}

Extension Lagrange's theorem can be extended to the equation of indices between three subgroups of G.

If we take K = {e} (e is the identity element of G), then [G : {e}] = |G| and [H : {e}] = |H|. Therefore, we can recover the original equation |G| = [G : H] |H|.

Applications 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

a n = e . {\displaystyle \displaystyle a^{n}=e{\mbox{.}}}

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, since the subgroup generated by any non-identity element must be the whole group itself. Lagrange's theorem can also be used to show that there are infinitely many primes: suppose there were a largest prime p {\displaystyle p} . Any prime divisor q {\displaystyle q} of the Mersenne number 2 p − 1 {\displaystyle 2^{p}-1} satisfies 2 p ≡ 1 ( mod q ) {\displaystyle 2^{p}\equiv 1{\pmod {q}}} (see modular arithmetic), meaning that the order of 2 {\displaystyle 2} in the multiplicative group ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} is p {\displaystyle p} . By Lagrange's theorem, the order of 2 {\displaystyle 2} must divide the order of ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} , which is q − 1 {\displaystyle q-1} . So p {\displaystyle p} divides q − 1 {\displaystyle q-1} , giving p < q {\displaystyle p<q} , contradicting the assumption that p {\displaystyle p} is the largest prime.

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