[代數] 兩個有關group的問題

[ssss50201 (ssss)]

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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◆ From: 108.3.154.49
→ smallray0705:a*b=a+b+ab 注意後面是實數運算..所以ae不等於a
→ smallray0705:上面少寫2倍..a=a*e=a+e+2ae => 0=(1+2a)e 剩下自己
→ Sumboy      :我覺得(ab)^2 = (ab)(ab)耶,似乎abelian是必要的