4.3: Exercises

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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.

Every group contains at least two distinct subgroups.
If \(H\) is a proper subgroup of group \(G\) and \(G\) is finite, then we must have \(|H|\lt |G|\text{.}\)
\(7\mathbb{Z}\) is a subgroup of \(14\mathbb{Z}\text{.}\)
A group \(G\) may have two distinct proper subgroups which are isomorphic (to one another).

2. Give specific, precise examples of the following groups \(G\) with subgroups \(H\text{:}\)

A group \(G\) with a proper subgroup \(H\) of \(G\) such that \(|H|=|G|\text{.}\)
A group \(G\) of order \(12\) containing a subgroup \(H\) with \(|H|=3\text{.}\)
A nonabelian group \(G\) containing a nontrivial abelian subgroup \(H\text{.}\)
A finite subgroup \(H\) of an infinite group \(G\text{.}\)

3. Let \(n\in \mathbb{Z}^+\text{.}\)

Prove that \(n\mathbb{Z} \leq \mathbb{Z}\text{.}\)
Prove that the set \(H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A=\pm 1\}\) is a subgroup of \(GL(n,\mathbb{R})\text{.}\)

(Note: Your proofs do not need to be long to be correct!)

4. Let \(n\in \mathbb{Z}^+\text{.}\) For each group \(G\) and subset \(H\text{,}\) decide whether or not \(H\) is a subgroup of \(G\text{.}\) In the cases in which \(H\) is not a subgroup of \(G\text{,}\) provide a proof. (Note. Your proofs do not need to be long to be correct!)

\(G=\mathbb{R}\text{,}\) \(H=\mathbb{Z}\)
\(G=\mathbb{Z}_{15}\text{,}\) \(H=\{0,5,10\}\)
\(G=\mathbb{Z}_{15}\text{,}\) \(H=\{0,4,8,12\}\)
\(G=\mathbb{C}\text{,}\) \(H=\mathbb{R}^*\)
\(G=\mathbb{C}^*\text{,}\) \(H=\{1,i,-1,-i\}\)
\(G=\mathbb{M}_n(\mathbb{R})\text{,}\) \(H=GL(n,\mathbb{R})\)
\(G=GL(n,\mathbb{R})\text{,}\) \(H=\{A\in \mathbb{M}_n(\mathbb{R})\,:\,\det A = -1\}\)

5. Let \(G\) and \(G'\) be groups, let \(\phi\) be a homomorphism from \(G\) to \(G'\text{,}\) and let \(H\) be a subgroup of \(G\text{.}\) Prove that \(\phi(H)\) is a subgroup of \(G'\text{.}\)

6. Let \(G\) be an abelian group, and let \(U=\{g\in G\,:\, g^{-1}=g\}.\) Prove that \(U\) is a subgroup of \(G\text{.}\)