Editing Lagrange's theorem (group theory) (section)

== Applications ==
A consequence of the theorem is that the [[order (group theory)|order of any element]] {{mvar|a}} of a finite group (i.e. the smallest positive integer number {{mvar|k}} with {{math|{{mvar|a}}<sup>{{mvar|k}}</sup> {{=}} {{mvar|e}}}}, where {{mvar|e}} is the identity element of the group) divides the order of that group, since the order of {{mvar|a}} is equal to the order of the [[cyclic group|cyclic]] subgroup [[generating set of a group|generated]] by {{mvar|a}}. If the group has {{mvar|n}} elements, it follows

:<math>\displaystyle a^n = e\mbox{.}</math>

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 group|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 <math>p</math>. Any prime divisor <math>q</math> of the [[Mersenne number]] <math>2^p -1</math> satisfies <math>2^p \equiv 1 \pmod {q}</math> (see [[modular arithmetic]]), meaning that the order of <math>2</math> in the [[multiplicative group]] <math>(\mathbb Z/q\mathbb Z)^*</math> is <math>p</math>. By Lagrange's theorem, the order of <math>2</math> must divide the order of <math>(\mathbb Z/q\mathbb Z)^*</math>, which is <math>q-1</math>. So <math>p</math> divides <math>q-1</math>, giving <math> p < q </math>, contradicting the assumption that <math>p</math> is the largest prime.<ref>{{Citation|last1=Aigner|first1=Martin|author-link=Martin Aigner|last2=Ziegler|first2=Günter M.|author2-link=Günter M. Ziegler|year=2018|title=[[Proofs from THE BOOK]]|chapter=Chapter 1|pages=3–8|publisher=Springer|location=Berlin|edition=Revised and enlarged sixth|isbn=978-3-662-57264-1}}</ref>