Abstract Algebra 65: Fundamental theorem of finite abelian groups, Part II

Abstract Algebra 65: Fundamental theorem of finite abelian groups, Part II

Abstract: We state and give examples for the fundamental theorem of finite abelian groups. This theorem states that every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. In particular, we discuss how Z/9Z and Z/3Z x Z/3Z are the only two abelian groups of size 9 up to isomorphism. We also list all the abelian groups of size 72 up to isomorphism.

This video accompanies the class "Introduction to Abstract Algebra" at Colorado State University:

[scorpion]

Thank you so much for all your hard work, you videos have been tremendously helpful! I just have a quick question, this sounds counterintuitive but I think what I just learned from this video is that any finite abelian group of let's say size 72 is isomorphic to products of cyclic groups (all 6 of them) but those 6 groups are not isomorphic to each other.
intuitively it would make sense if the abelian group of size 72 would only be isomorphic to Z/8Z x Z/9Z because 8 and 9 are relatively prime.
Also did we not learn if A is isomorphic to B and B is isomorphic to C then A is also isomorphic to C? in here we have an abelian group of size 72 being isomorphic to Z/8Z x Z/9Z and also the same abelian group is isomorphic to Z/2Z x Z/4Z x Z/9Z for example, but Z/8Z x Z/9Z is not isomorphic to Z/2Z x Z/4Z x Z/9Z because the orders are different so how is this possible that the abelian group would be isomorphic to them both? Won't the abelian group of size 72 must have elements of certain orders in which it would be different than some of these products?
I am a little confused on this part and apologize for writing so much lol. Thanks in advance.