- #1
davidbenari
- 466
- 18
My professor says a homomorphism is an association between groups such that multiplication tables are preserved (he says nothing else, this is his definition).
Imagine two groups ##G## and ##H## with ##|G|=2## and ##|H|=8##. In his way of seeing things, there could be a homomorphism ##\mu## that associates elements of G to collections of elements of ##H##. The collections must satisfy the same multiplication table.
But I don't agree with such a definition. The definition math textbooks give is that of a mapping ##\mu: G \to H## which follows the multiplicative identity ##\mu(gk)=\mu(g) \mu(k)##. His definition is not a mapping at all, since mapping can't be 1 to n, and even if you relax that condition, a homomorphism maps to an element not to a collection.
So where is he getting these notions from? Anyone agrees with him? Where can I read a definition which is OK with his idea? If he's wrong, could you tell me why do you think so?
edit: My professor has been wrong before, so don't think I'm taking false words out of his mouth. That said, if you could give me a sketch on important points on how he's wrong it could be helpful. I've corrected him before and its getting on my nerves that I'm being taught material that's bogus. The other thing though, is that this professor is an important researcher, so he's definitely not dumb; so maybe something complicated is going on.
Imagine two groups ##G## and ##H## with ##|G|=2## and ##|H|=8##. In his way of seeing things, there could be a homomorphism ##\mu## that associates elements of G to collections of elements of ##H##. The collections must satisfy the same multiplication table.
But I don't agree with such a definition. The definition math textbooks give is that of a mapping ##\mu: G \to H## which follows the multiplicative identity ##\mu(gk)=\mu(g) \mu(k)##. His definition is not a mapping at all, since mapping can't be 1 to n, and even if you relax that condition, a homomorphism maps to an element not to a collection.
So where is he getting these notions from? Anyone agrees with him? Where can I read a definition which is OK with his idea? If he's wrong, could you tell me why do you think so?
edit: My professor has been wrong before, so don't think I'm taking false words out of his mouth. That said, if you could give me a sketch on important points on how he's wrong it could be helpful. I've corrected him before and its getting on my nerves that I'm being taught material that's bogus. The other thing though, is that this professor is an important researcher, so he's definitely not dumb; so maybe something complicated is going on.
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