s*******3
发帖数: 17 | 1 let f:G-G' be a group homomorphism. show that if [G] is finite, then [f(G)]
is finite and is a divisor of [G]
show that any group homomorphism f:G-G' where [G] is a prime must either be
the trivial homomorphism or a one-to-one map
我觉得第二道可以用第一个道的结论直接做啊,但是还是后面不会证one-to-one 那步
求教了!谢谢谢谢 | r******n
发帖数: 149 | 2 这个不是简单利用同态基本定理么
1.
G/Kerf = Im(f)
So
[f(G)]=[Im(f)] and since
[G]/[Kerf]=[Im(f)]
we have [G]=[Im(f)][Kerf]
2. if [G] is prime, then
[f(G)]=1 => ker f = G => trivial
or [f(G)]=G => one to one.
好久没接触这些了，不知道对不对，呵呵
be
【在 s*******3 的大作中提到】
: let f:G-G' be a group homomorphism. show that if [G] is finite, then [f(G)]
: is finite and is a divisor of [G]
: show that any group homomorphism f:G-G' where [G] is a prime must either be
: the trivial homomorphism or a one-to-one map
: 我觉得第二道可以用第一个道的结论直接做啊,但是还是后面不会证one-to-one 那步
: 求教了!谢谢谢谢
| r*****d
发帖数: 57 | 3 素数的因子只有1和它自己，分别对应平凡映射（像为{0}，即核为原像全体）和1-1映
射，即核为{0}。 |
|
