Cauchy's Theorem
If is prime and for finite , then some has order .
Theorem 4.13 (Cauchy's Theorem)
If and is a prime that divides , then there is such that .
Proof
Let be the set of -tuples whose product is . The cyclic group acts by cyclic rotation. There are such tuples, and each orbit has size or by orbit-stabiliser. The number of orbits of size is divisible by , while gives . Hence another constant tuple occurs with . It belongs to , so , and therefore .
Related
Stated in
- Theorem 4.13 (Cauchy's Theorem)§4.2 Orbit-Stabiliser Theorem