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Solve: Onto Homomorphisms

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1. Let R* be the set of nonzero real numbers. Then R* is a group with the usual multiplication.
a) Let {}: R* --> R* be defined by {}(x) = x^-3, for any x is a member of R*. Show that {} is a group isomorphism.
b) Let theta: R* --> R* be defined by theta(x) = 2^x, for any x is a member of R*. Show that theta is a group homomorphism. Is theta an isomorphism? Why?
c) Let psi: R* --> R* be defined by psi(x) = 3x, for any x a member of R*. Is psi a homomorphism? Why?

a) Show that varphi: Z_2 --> Z_2 defined by varphi(x) = x^2 is a homomorphism.
b) Show that {}: Z_3 --> Z_3 defined by {}(x) = x^3 is a homomorphism.
c) Show that theta: Z_4 --> Z_4 defined by theta(x) = x^4 is not a homomorphism.
d) What can you say about psi: Z_n --- Z_n defined by varphi(x) = x^n?

3. Prove that Z_12/ < 4 > ~ = Z_4 by first constructing an onto homomorphism from from Z_12 to Z_4 with the subgroup < 4 > as its kernel, then apply the First Isomorphism Theorem.

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Solution Summary

This solution shows calculations formatted in the attached Word document. Note: Some of the problems are wrong and I have replaced homomorphism with isomorphism where necessary. All required steps are provided.