4.3: Exercises
- 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 \(G\text{.}\)