Cauchy theorem: a good place to review elementary group theory stuff

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 p-group results, including Sylow’s theorems.

Theorem (Cauchy):

G a finite group and p a prime. Then if p\big| |G| \Rightarrow \exists g\in G s.t. g‘s order is p.

Proof:

(i)When G is abelian: do induction on |G|. For |G|=n: consider a \in G, a\neq e. If p\big |\langle a\rangle|, done; otherwise p\big| [G:\langle a\rangle], by inductive hypothesis \exists x\langle a\rangle of order p in G/\langle a\rangle for some x\in G. If m is the order of x in G, then (x\langle a\rangle)^m=\langle a\rangle\Rightarrow p|m, done.

(ii)For a general G: also do induction on |G|. for |G|=n: Z(G) is abelian. If p\big| |Z(G)|, done by (i); otherwise from conjugacy class equation, \exists \text{Cl}(a), where a\notin Z(G), s.t. p\nmid |\text{Cl}(a)|=[G:C_G(a)]\Rightarrow p\big||C_G(a)|<n=|G|, done by induction.

 

 

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