Can someone give me a hand on this problem? I want to show that for all . Given that are abelian groups such that
Suppose . We have and .
But we know each of and can be decomposed into a product of cyclic groups. So, where is a cyclic group of order x.
So, I try to show that
But I don't really know how to apply the hypothesis to prove this here.
Here is my attempt to show one direction. Can someone help me if I'm on the right track.
Instead of decomposing each and into direct products of cyclic subgroups, I tried to do induction on
I have and
For , then and .
By hypothesis, , so we can use transitivity of group isomorphism to get
Assume this is true for , i.e for .
I want to show this is true for
Here can I use the argument that if where are finite abelian, then to say that ? I appreciate any help.