what's the differece between linear permutations and isomorphisms over a finite group?

Question

I read two concepts in book. One is linear permutation, the other is isomorphism. I found they almost the same in the follow context.

Let $(A,+)$ and $(B,+)$ be finite abelian groups. Let $f$ be a linear permutation from $(A,+)$ to $(B, +)$, and $g$ be a group isomorphism from $(A,+)$ to $(B, +)$.

$f$ and $g$ both have the same definition domain and codomain satisifying the conditions:

1. $f(0)=0$, $g(0)=0$;
2. For any $a,b \in A$, $f(a+b)=f(a)+f(b)$, $g(a+b)=g(a)+g(b)$;
3. $f$ and $g$ are bijective.

I'm confused that $f$ and $g$ are the same things. Would anyone tell me the difference between them ? Giving examples is referred. Thanks.

Answer 1

An isomorphism is a bijection, which can be viewed as a permutation.

An isomorphism is also a homomorphism, which gives [what I assume is] the "linear" in "linear permutation."

As far as I can tell, these are two different names for the same thing (in this context).