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# Group Theory : Homomorphisms, Kernels, Isomorphisms and Fields

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(a) If G1 and G2 are groups, define what it means to describe a function
h:G1 -> G2 as a homomorphism.

(b) If h: G1 ?> G2 is a homomorphism, define the kernel of h.Prove that the range of h is a subgroup of G2 , and that the kernel of h is a normal subgroup of G1.

(c) Let G be the group of 2x2 real matrices under addition. Prove that precisely two of the three functions below is a homomorphism. Identify the kernel and range of each of those that is a homomorphism.

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(d)For the two homomorphisms in (c), prove that the two quotient groups
G/N, where N is the corresponding kernel, are isomorphic.

6 (a) Let S ={...}. Prove that, under the usual operations of addition and multiplication, S is a field.
b) Let T={a+ b sqrt 3| a,b E Z}.
List the elements of T.
Addition and multiplication are defined on T in the natural way, with the coefficients being reduced modulo 3, as necessary. Prove that T is a commutative ring with a one.
By considering the element sqrt 3 prove that T is not a field.
Is T an integral domain?