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Reading Questions 11.3 Reading Questions

1.

Consider the function \(\phi:\mathbb Z_{10}\rightarrow\mathbb Z_{10}\) defined by \(\phi(x)=x+x\text{.}\) Prove that \(\phi\) is a group homomorphism.

2.

For \(\phi\) defined in the previous question, explain why \(\phi\) is not a group isomorphism.

3.

Compare and contrast isomorphisms and homomorphisms.

4.

Paraphrase the First Isomorphism Theorem using only words. No symbols allowed at all.

5.

“For every normal subgroup there is a homomorphism, and for every homomorphism there is a normal subgroup.” Explain the (precise) basis for this (vague) statement.