1-3) determine whether the given map (theta) is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not?(adsbygoogle = window.adsbygoogle || []).push({});

1) <Z,+> with <Z,+> where theta(n)= -n for n elements in Z

2) <Z,+> with <Z,+> where theta(n)= 2n for n elements in Z

3) <Z,+> with <Z,+> where theta(n)= n+1 for n elements in Z

4) Let alpha: <Z4,+> -> <Z#5,X> by alpha(0)=1, alpha(1)=2, alpha(2)=4, alpha(3)=3. Prove alphais an isomorphism of groups. (verify 16 equations).

5) Let gamma:<C*,X> -> <R+,X> by gamma(a+bi)= a^2 + b^2. Prove gamma is a homomorphism.

6) Let theta: <Z,+> -> <{+/-1,+/-1},X> by theta(k)=i^k. Prove theta is homomorpism.

7) Let G be a group. Prove that alpha:G->G by alpha(a)=a^2 is a homomorphism if and only if G is abelian.

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# 7 iso and hom questions

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