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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.
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If there exists a homomorphism \(\phi\,:\,G\to G'\text{,}\) then \(G\) and \(G'\) must be isomorphic groups.
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There is an integer \(n\geq 2\) such that \(\mathbb{Z}\simeq \mathbb{Z}_n\text{.}\)
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If \(|G|=|G'|=3\text{,}\) then we must have \(G\simeq G'\text{.}\)
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If \(|G|=|G'|=4\text{,}\) then we must have \(G\simeq G'\text{.}\)
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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!)
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The function \(f:\mathbb{Z}\to\mathbb{Z}\) defined by \(f(n)=2n\text{.}\)
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The function \(g:\mathbb{R}\to\mathbb{R}\) defined by \(g(x)=x^2\text{.}\)
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The function \(h:\mathbb{Q}^*\to\mathbb{Q}^*\) defined by \(h(x)=x^2\text{.}\)
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Define \(d : GL(2,\mathbb{R})\to \mathbb{R}^*\) by \(d(A)=\det A\text{.}\) Prove/disprove that \(d\) is:
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a homomorphism
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1-1
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onto
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an isomorphism.
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Complete the group tables for \(\mathbb{Z}_4\) and \(\mathbb{Z}_8^{\times}\text{.}\) Use the group tables to decide whether or not \(\mathbb{Z}_4\) and \(\mathbb{Z}_8^{\times}\) are isomorphic to one another. (You do not need to provide a proof.)
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Let \(n\in \mathbb{Z}^+\text{.}\) Prove that \(\langle n\mathbb{Z},+\rangle \simeq \langle \mathbb{Z},+\rangle\text{.}\)
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Let \(G\) and \(G'\) be groups, where \(G\) is abelian and \(G\simeq G'\text{.}\) Prove that \(G'\) is abelian.
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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.
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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 \(\phi(a)^{-1}=\phi(a^{-1})\text{.}\)