shadi's profile沙地PhotosBlogListsMore  |  Tools Help |
|
2/20/2007 original question3. Let G be a finite set and * an associative binary operation on G with identity element e.
Suppose that G satisfies the cancellations laws, i.e.( f*g = f*h => g=h) and (g*f = h*f => g=h)
Prove that G is a group.
the solution given by the school goes:
Let G have n elements, say G={g1,...,gn}. Let f belongs to G and consider the set { f*g1, f*g2, ... , f*gn}. by the cancellation law, all the elements f*gi are distinct and there are n of them; hence { f*g1, f*g2, ... , f*gn} = G. Thus one of the f*gi = e. Similarly there exist j such that gj*f= e .
Now gj*f*gi = gj*e = gj and also gj*f*gi = e*gi = gi (using associativity) and hence gi= gj is the inverse of f.
Thus every element has an inverse and so G is a group.
It's not a difficult q, but the answer school gives out, which is too simple and inappropriate not satisfies me at all.
Just saying ,or maybe, this is the only question I've seriously taking comparison. Just a coincidence or am I wrong?
Even if I could it been all be gray,
but your picture on my wall,
it reminds me that it's not so bad,
it's not so bad at all.
p.s. 特别感谢樊同学的关注. Comments (9)
TrackbacksThe trackback URL for this entry is: http://greenlandwsd.spaces.live.com/blog/cns!FBF6FBB346E28AC8!739.trak Weblogs that reference this entry
|
|
|
