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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)



True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let \(G\) and \(G'\) be groups.

  1. If there exists a homomorphism \(\phi\,:\,G\to G'\text{,}\) then \(G\) and \(G'\) must be isomorphic groups.

  2. There is an integer \(n\geq 2\) such that \(\Z\simeq \Z_n\text{.}\)

  3. If \(|G|=|G'|=3\text{,}\) then we must have \(G\simeq G'\text{.}\)

  4. If \(|G|=|G'|=4\text{,}\) then we must have \(G\simeq G'\text{.}\)


For each of the following functions, prove or disprove that the function is (i) a homomorphism; (ii) an isomorphism. (Remember to work with the default operation on each of these groups!)

  1. The function \(f:\Z\to\Z\) defined by \(f(n)=2n\text{.}\)

  2. The function \(g:\R\to\R\) defined by \(g(x)=x^2\text{.}\)

  3. The function \(h:\Q^*\to\Q^*\) defined by \(h(x)=x^2\text{.}\)


LDefine \(d : GL(2,\R)\to \R^*\) by \(d(A)=\det A\text{.}\) Prove/disprove that \(d\) is:

  1. a homomorphism

  2. 1-1

  3. onto

  4. an isomorphism.


Complete the group tables for \(\Z_4\) and \(\Z_8^{\times}\text{.}\) Use the group tables to decide whether or not \(\Z_4\) and \(\Z_8^{\times}\) are isomorphic to one another. (You do not need to provide a proof.)


Let \(n\in \Z^+\text{.}\) Prove that \(\langle n\Z,+\rangle \simeq \langle \Z,+\rangle\text{.}\)


  1. Let \(G\) and \(G'\) be groups, where \(G\) is abelian and \(G\simeq G'\text{.}\) Prove that \(G'\) is abelian.

  2. Give an example of groups \(G\) and \(G'\text{,}\) where \(G\) is abelian and there exists a homomorphism from \(G\) to \(G'\text{,}\) but \(G'\) is NOT abelian.


Let \(\langle G,\cdot\rangle\) and \(\langle G',\cdot'\rangle\) be groups with identity elements \(e\) and \(e'\text{,}\) respectively, and let \(\phi\) be a homomorphism from \(G\) to \(G'\text{.}\) Let \(a\in G\text{.}\) Prove that \(\phi(a)^{-1}=\phi(a^{-1})\text{.}\)