proof of Cauchy’s theorem

Let G be a finite groupMathworldPlanetmath, and suppose p is a prime divisorPlanetmathPlanetmath of |G|. Consider the set X of all p-tuples (x1,,xp) for which x1xp=1. Note that |X|=|G|p-1 is a multipleMathworldPlanetmathPlanetmath of p. There is a natural group actionMathworldPlanetmath of the cyclic groupMathworldPlanetmath /p on X under which m/p sends the tuple (x1,,xp) to (xm+1,,xp,x1,,xm). By the Orbit-Stabilizer Theorem, each orbit contains exactly 1 or p tuples. Since (1,,1) has an orbit of cardinality 1, and the orbits partitionMathworldPlanetmathPlanetmath X, the cardinality of which is divisible by p, there must exist at least one other tuple (x1,,xp) which is left fixed by every element of /p. For this tuple we have x1==xp, and so x1p=x1xp=1, and x1 is therefore an element of order p.


  • 1 James H. McKay. Another Proof of Cauchy’s Group Theorem, American Math. Monthly, 66 (1959), p119.
Title proof of Cauchy’s theorem
Canonical name ProofOfCauchysTheorem
Date of creation 2013-03-22 12:23:30
Last modified on 2013-03-22 12:23:30
Owner yark (2760)
Last modified by yark (2760)
Numerical id 8
Author yark (2760)
Entry type Proof
Classification msc 20E07
Classification msc 20D99