Abstract algebra [Lecture notes] by Thomas C. Craven PDF

By Thomas C. Craven

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Why do we expect that normality is the condition we need to define the operation on cosets? For one thing, it makes things look closer to abelian. For another, we expect, from ring theory, that we should get quotient groups as sets of cosets, and they should be images of homomorphisms where the kernels are the subgroups we are factoring out. But the kernel of a homomorphism must be a normal subgroup: To see this, we first define kernel, just as we did for rings: for a group homomorphism f : G → H, ker f = { g ∈ G | f (g) = eH }.

While they are quite true, the author could have simplified the results further: x2 + x + 1 ≡ 1 (mod x + 1) and 3x4 + 4x2 + 2x + 2 ≡ 2x + 1 (mod x2 + 1). The former reduction can be gotten by simply thinking of x ≡ −1 (mod x + 1) and the latter by thinking of x2 ≡ −1 (mod x2 + 1). To argue that such manipulations are legal requires the next two theorems, whose proofs are the same as the corresponding theorems for the integers. 1). Congruence modulo p(x) is an equivalence relation on the set F [x].

We often write it as + if the group is abelian, because it then behaves like addition in a ring. In fact, any ring is a group if we consider only its + operation. If the group is nonabelian, a more common notation is the one we use for multiplication: ab for a ∗ b. In the former case, we write inverses as −a and e as 0. In the latter we write inverses as a−1 , but e is still the usual name for the identity. Examples: we have already seen essentially all the abelian finite groups. 7 says they are just the additive subgroups of our rings Zn .

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Abstract algebra [Lecture notes] by Thomas C. Craven

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