2 2 2
若{G,*}為一群，若且唯若(a * b) = a * b
所有的a,b屬於R , 則 G為交換群，試證之。
(P.F)
2 2 2
→: (a * b) = a * b => a * b * b * a
2 2 2
因為 (a * b) = (a * b) * (a * b) = a * b
-1 -1 -1 2 2 -1
=>a * a * b * a * b * b = a * a * b * b
=>b * a = a * b 為交換群
------------------------------------------------(1)
2 2 2
←: G 為交換群 => (a * b ) = a * b
(a * b) = (a * b) * (a * b) = a * (b * a) * b
2 2
= a * (a * b) * b = (a * a) * (b * b) = a * b
------------------------------------------------(2)
我的向右證明是
-1
因為為群 let b = a
2 2 2
(a * a ^ -1) = a * (a ^ -1) = e * e = e
2 2 2
(a ^ -1 * a) = (a ^ -1) * a = e * e = e
-1 -1
所以 a * a = a * a = e 所以為交換群
------------------------------------------------
以上(1)的證明我看不懂，可以幫忙解釋嗎?
跟我的證明這樣可以嗎?
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