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.
