[代數] cyclic group

[mqazz1 (無法顯示)]

1. Let A = {x, a, b, c, d}

        (a) how many closed binary operations f on A satisfy f(a,b)=c

                這個我知道是5^24

        (b) how many of the functions f in part(a) have an identity?

                3 * 5^15

        (c) how many of the functions f in part(a) are commutative?

                3 * 5^9

        請問(b)跟(c)是為什麼?

2. Find all the elements of order 10 in (Z_40, +)

        請問這題要怎麼找?

謝謝

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推 Minkowski   :f has an identity? 什麼意思?
→ mqazz1      :我是照題目打的 http://ppt.cc/e6vY
→ keroro321   :2. 根據條件知道 a≠1, b≠1
→ keroro321   :B)類似像這樣假設x是1 (a,?)=3,(b,?)=(c,?)=(d,?)=4
→ keroro321   :x,c,d都有可能所以乘3
→ keroro321   :f(x,s)=f(s,x)=s ,where s=a,b,c,d,x <=> x is an ID
→ keroro321   :因為f是函數所以f至多只有一個identity.(沒說明到)