proof of Cauchy’s theorem
Let be a finite group, and suppose is a prime divisor of . Consider the set of all -tuples for which . Note that is a multiple of . There is a natural group action of the cyclic group on under which sends the tuple to . By the Orbit-Stabilizer Theorem, each orbit contains exactly or tuples. Since has an orbit of cardinality , and the orbits partition , the cardinality of which is divisible by , there must exist at least one other tuple which is left fixed by every element of . For this tuple we have , and so , and is therefore an element of order .
- 1 James H. McKay. Another Proof of Cauchy’s Group Theorem, American Math. Monthly, 66 (1959), p119.
|Title||proof of Cauchy’s theorem|
|Date of creation||2013-03-22 12:23:30|
|Last modified on||2013-03-22 12:23:30|
|Last modified by||yark (2760)|