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Contents Index Dark Mode Prev Up Next \(\newcommand{\identity}{\mathrm{id}}
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\)
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.
