characterization of full families of groups
Proof. ,,” Assume that is a nontrivial group homomorphism. Then define
as follows: if is such that and is such that , then . If is such that , then and for . This values uniquely define and one can easily check that is not decomposable.
,,” Assume that for any such that we have that any homomorphism is trivial. Let
be any homomorphism. Moreover, let and be such that . We wish to show that .
So assume that . Then there exists such that . Let
be the projection and let
Corollary. Assume that is a family of nontrivial groups such that is periodic for each . Moreover assume that for any such that and any , orders and are realitvely prime (which implies that is countable). Then is full.
Proof. Assume that and is a group homomorphism. Then divides for any . But , so and are relatively prime. Thus , so . Therefore is trivial, which (due to proposition) completes the proof.
|Title||characterization of full families of groups|
|Date of creation||2013-03-22 18:36:08|
|Last modified on||2013-03-22 18:36:08|
|Last modified by||joking (16130)|