|
原帖由 也和話 於 2009-1-11 11:12 發表 [一道非常宗教性的數學題 - 信仰天地 - backchina.com]
Given a set W and a binary operation #, let (W, #) be a group. For any element x, y in W and any fixed element s in W, define the binary operation @ as x@y = x#s#y.
Show that (W, @) is a group.
Show that (W, @) is isomorphic to (W, #).
Express x#y using the binary operation @.
1. It is trivial to show that (W, @) is a group.
2. Define f from (W, @) to (W, #) as f(x) = x#s. It can be shown that f (x@y) = f(x) # f(y).
so (W, @) is isomorphic to (W, #).
3. Denote E the identity element in the group (W, #), and iE the inverse element of E in the group (W, @),
Then you can express # in terms of @ as following: X # Y = X @ iE @ Y.
To "問?": (-s)#(-s) = (-s)#E#(-s) = iE. The operation of # is still needed, but you just don't see it this way. |
|