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Embed Dark Mode Prev Up Next \(\require{mathtools}\require{color} \setcounter{MaxMatrixCols}{15}
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Section Group Homomorphisms
Worksheet Part 1: Homomorphisms and the First Isomorphism Theorem
The goal of these problems is to explore group homomorphisms and their properties and practice using the First Isomorphism Theorem.
1. Let
\(\R^*\) be the group of nonzero real numbers under multiplication and
\(r\) be a positive integer. Show that the map that takes
\(x\) to
\(x^r\) is a homomorphism from
\(\R^*\) to
\(\R^*\) and determine the kernel.
2.
In this problem you will determine all possible homomorphisms
\(\phi\) from
\(\Z_{12}\) to
\(\Z_{30}\text{.}\)
(a) Use the properties of homomorphisms to explain why such a
\(\phi\) is totally determined by
\(\phi(1)\text{.}\)
(b)
(c) Explicitly list all of the homomorphisms from
\(\Z_{12}\) to
\(\Z_{30}\text{.}\)
3. Show that the map
\(\pi_G: G\times H \to G\) given by
\(\pi_G(g,h)=g\) is a homomorphism (called the
projection of
\(G\times H\) onto
\(G\) ). What is the kernel of this map?
4. Prove that
\((G\times H)/(\{e\}\times H)\isom G\text{.}\)
5. Show that the map
\(\phi: \Z\times \Z \to \Z\) given by
\(\phi(a,b)=a-b\) is a homomorphism. What is the kernel of this map?
6. Prove that
\((\Z \times \Z)/\subgroup{(1,1)}\isom \Z\text{.}\)
7. Let
\(G\) be an Abelian group,
\(G_n=\{g \mid g^n=e\}\text{,}\) and
\(G^n=\{g^n \mid g \in G\}\text{.}\) (
\(G_n\) is the subgroup of all elements with order dividing
\(n\) and
\(G^n\) is the subgroup of all
\(n\) -th powers.)
Show that
\(G/G_n \isom G^n\text{.}\) Why is it important that
\(G\) is Abelian in this problem?
