decomposable homomorphisms and full families of groups
Let be two families of groups (indexed with the same set ).
Definition. We will say that a homomorphism
is decomposable if there exists a family of homomorphisms such that
Remarks. For each and we will say that if for any . One can easily show that any homomorphism
is decomposable if and only if for any and any such that we have . This implies that if is an isomorphism and is decomposable, then each homomorphism in decomposition is an isomorphism and
Also it is worthy to note that composition of two decomposable homomorphisms is also decomposable and
Definition. We will say that family of groups is full if each homomorphism
Remark. It is easy to see that if is a full family of groups and , then is also a full family of groups.
Example. Let . Then is full. Indeed, let
be a group homomorphism. Then, for any and such that we have that divides and thus divides , so it is easy to see that . Therefore (due to first remark) is decomposable.
Counterexample. Let be two copies of . Then is not full. Indeed, let
be a group homomorphism defined by
Now assume that . Then we have:
|Title||decomposable homomorphisms and full families of groups|
|Date of creation||2013-03-22 18:36:03|
|Last modified on||2013-03-22 18:36:03|
|Last modified by||joking (16130)|