Re: [理工] [離散]cyclic group 必與 (Z , +) 或 ( …

[mqazz1 (無法顯示)]

※ 引述《da0910cc (da0910cc)》之銘言：
: 請問標題那個概念
: 要怎麼想呢?
                                 k
先令cyclic group是G, G = <a> = {a  | k為整數}


case 1: G是infinite group

                                 i
   定義f: (G, *) -> (Z, +) 為 f(a ) = i

           i   j     i+j           i     j
        f(a * a ) = a   = i+j = f(a )+f(a )             => f為homomorphism

             i     j                 i   j
        若f(a )=f(a )  =>  i=j  =>  a = a               => f為1-1

                             i               i
        對於所有得i屬於Z, 取a 屬於G, 使得 f(a )=i       => f為onte

   所以 G和(Z,+)同構



case 2: G是finite grouop, 假設o(G)=n

                         2        n-1
    令 G = <a> = {e, a, a , ..., a   }

                        m
    定義f: G -> Zn 為f(a )=[m], 0 <= m <= n-1

           i  j       i+j                         i     j
        f(a *a ) = f(a   ) = [i+j] = [i]+[j] = f(a )+f(a ) => f為homomorphism

        再證f為1-1和onto

    所以(G, *)和(Zn, +)同構

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