[代數] 兩個有關group的問題
我覺得學group好痛苦阿
遇到判斷題幾乎都是死翹翹>"<
1. Let * be the binary operation on the rational numbers is given by
a*b=a+b+2ab. Which of the following are true?
(A) * is commutative
(B) There is a rational number that is a *-identity
(C) Every rational number has a *-inverse.
Ans: A,B
我知道A是對的（因為Q本身就commutative）
關於B，我試著用a*e=a解identity e
a*e=a+e+2ae=a+e+2a=2a+e = a (by assumption e is identity, a*e=a)
所以e=-a
但當我想檢查我找到的e是不是能讓a*e=e時，卻不對了
關於C, 沒有頭緒怎麼找inverse在沒有找到identity之前...
2. A group G in which (ab)^2=(a^2)(b^2), for all a, b in G is necessarily
(A) finite
(B) cyclic
(C) of order 2
(D) abelian
(E) none of the above
Ans:D
我有嘗試看看是不是abilian
(ba)^2=(b^2)(a^2) =? (ab)^2 不知道怎麼從=?左邊推到右邊
謝謝幫忙~~
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