proof of Fermat’s little theorem using Lagrange’s theorem


If a,pZ with p a prime and pa, then ap-11(modp).


We will make use of Lagrange’s Theorem: Let G be a finite groupMathworldPlanetmath and let H be a subgroupMathworldPlanetmathPlanetmath of G. Then the order of H divides the order of G.

Let G=(/p)× and let H be the multiplicative subgroup of G generated by a (so H={1,a,a2,}). Notice that the order of H, h=|H| is also the order of a, i.e. the smallest natural numberMathworldPlanetmath n>1 such that an is the identityPlanetmathPlanetmathPlanetmath in G, i.e. ah1modp.

By Lagrange’s theorem h|G|=p-1, so p-1=hm for some m. Thus:


as claimed. ∎

Title proof of Fermat’s little theorem using Lagrange’s theorem
Canonical name ProofOfFermatsLittleTheoremUsingLagrangesTheorem
Date of creation 2013-03-22 14:23:53
Last modified on 2013-03-22 14:23:53
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Proof
Classification msc 11-00
Related topic LagrangesTheorem