※ 引述《mqazz1 (無法顯示)》之銘言：
: Let G be a cyclic group
: If |G| = n, then G is isomorphic to (Zn, +)
: 請問這個式子該怎麼證呢?
: 謝謝
G be a cyclic group,假設g是G的生成員 => <g>=G
且 |G| = n => g^n=1
考慮 ψ: G → Zn,by ψ(g^a)=a (mod n)
then
ψ is one to one(=>) and well defined(<=)
g^a=g^b <=> g^(a-b)=1
<=> n|(a-b)
<=> a-b=0 (mod n)
<=> a=b (mod n)
ψ is onto
forall a屬於Zn => ψ(g^a)=a
ψ is homomorphic
forall a,b屬於Zn => ψ(g^a)ψ(g^b)=a+b=ψ(g^(a+b))
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 125.224.185.19