抽象代数,证明不存在一个群,里面有且只有两个二阶元(原题：Showthattheredoesnotexistagroupthathasexactly2elementsoforder2)

Question

抽象代数,证明不存在一个群,里面有且只有两个二阶元(原题：Show that there does not exist a group that has exactly 2 elements of order2)

Answer 1

设群G的单位元为e, 而a, b是G中两个二阶元(a ≠ b),
只需证明G中存在与a, b都不相等的二阶元c. 分两种情况:
1) 若ab = ba, 取c = ab.
∵aa = e, a ≠ b, ∴c ≠ e (否则a = ac = aab = b, 矛盾).
∵bb = e, a ≠ e, ∴c ≠ b (否则a = abb = cb = bb = e, 矛盾).
∵aa = e, b ≠ e, ∴c ≠ a (否则b = aab = ac = aa = e, 矛盾).
而∵aa = bb = e, ∴cc = (ab)(ab) = a(ba)b = a(ab)b = aabb = e.
综上, c是与a, b都不相等的二阶元.
2) 若ab ≠ ba, 取c = aba.
∵aa = e, a ≠ b, ∴ab ≠ e (否则a = aab = b, 矛盾),
∴c ≠ a (否则ab = abaa = ca = aa = e, 矛盾).
∵aa = e, ∴c ≠ b (否则ab = abaa = ca = ba, 矛盾).
∵aa = e, b ≠ e, ∴c ≠ e (否则b = aabaa = aca = aa = e, 矛盾).
而∵aa = bb = e, ∴cc = (aba)(aba) = ab(aa)ba = a(bb)a = aa = e.
综上, c是与a, b都不相等的二阶元.