Notice that we did not prove that H itself is abelian—only the image. This foreshadows the concept of a homomorphic image preserving certain properties but not all.
Pinter writes as if he is speaking to you. He uses second-person narrative. He anticipates your confusion. He tells you why a definition is chosen before he states it.
"Since G is abelian, ab=ba. Then f(ab)=f(a)f(b)=f(b)f(a)=f(ba). Hence f(G) is abelian." This is technically correct but pedagogically useless. It jumps from f(ab) to the conclusion without explaining why the image group inherits commutativity.
This is the book’s crown jewel. Pinter’s exercises are not computational drills. They are miniature explorations. He often asks you to discover a theorem before it is formally named. For example, he might ask: "Prove that in any group, the identity element is unique." You prove it. Then, in the next paragraph, he says, "The result you just proved is known as the Uniqueness of the Identity Theorem."
Since x and y are in f(G), there exist a, b in G such that f(a)=x and f(b)=y.
Notice that we did not prove that H itself is abelian—only the image. This foreshadows the concept of a homomorphic image preserving certain properties but not all.
Pinter writes as if he is speaking to you. He uses second-person narrative. He anticipates your confusion. He tells you why a definition is chosen before he states it.
"Since G is abelian, ab=ba. Then f(ab)=f(a)f(b)=f(b)f(a)=f(ba). Hence f(G) is abelian." This is technically correct but pedagogically useless. It jumps from f(ab) to the conclusion without explaining why the image group inherits commutativity.
This is the book’s crown jewel. Pinter’s exercises are not computational drills. They are miniature explorations. He often asks you to discover a theorem before it is formally named. For example, he might ask: "Prove that in any group, the identity element is unique." You prove it. Then, in the next paragraph, he says, "The result you just proved is known as the Uniqueness of the Identity Theorem."
Since x and y are in f(G), there exist a, b in G such that f(a)=x and f(b)=y.
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