Cauchy’s theorem in group theory is an excellent place to review fundamental group stuff because (one of) its proof(s) involves the concepts of normal group, centralizer, orbit-stabilizer theorem and conjugacy class equation, etc. On the other hand, it is a crucial step to many important -group results, including Sylow’s theorems.
a finite group and a prime. Then if s.t. ‘s order is .
(i)When is abelian: do induction on . For : consider , . If , done; otherwise , by inductive hypothesis of order in for some . If is the order of in , then , done.
(ii)For a general : also do induction on . for : is abelian. If , done by (i); otherwise from conjugacy class equation, , where , s.t. , done by induction.