5.3: Exercises
1. 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 \(G\) be a group with identity element \(e\text{.}\)
- If \(G\) is infinite and cyclic, then \(G\) must have infinitely many generators.
- There may be two distinct elements \(a\) and \(b\) of a group \(G\) with \(\langle a\rangle =\langle b\rangle\text{.}\)
- If \(a,b\in G\) and \(a\in \langle b\rangle\) then we must have \(b\in \langle a\rangle\text{.}\)
- If \(a\in G\) with \(a^4=e\text{,}\) then \(o(a)\) must equal \(4\text{.}\)
- If \(G\) is countable then \(G\) must be cyclic.
2. Give examples of the following.
- An infinite noncyclic group \(G\) containing an infinite cyclic subgroup \(H\text{.}\)
- An infinite noncyclic group \(G\) containing a finite nontrivial cyclic subgroup \(H\text{.}\)
- A cyclic group \(G\) containing exactly \(20\) elements.
- A nontrivial cyclic group \(G\) whose elements are all matrices.
- A noncyclic group \(G\) such that every proper subgroup of \(G\) is cyclic.
3. Find the orders of the following elements in the given groups.
- \(2\in \mathbb{Z}\)
- \(-i\in \mathbb{C}^*\)
- \(-I_2\in GL(2,\mathbb{R})\)
- \(-I_2\in \mathbb{M}_2(\mathbb{R})\)
- \((6,8)\in \mathbb{Z}_{10}\times \mathbb{Z}_{10}\)
4. For each of the following, if the group is cyclic, list all of its generators. If the group is not cyclic, write NC.
- \(5\mathbb{Z}\)
- \(\mathbb{Z}_{18}\)
- \(\mathbb{R}\)
- \(\langle \pi\rangle\) in \(\mathbb{R}\)
- \(\mathbb{Z}_2^2\)
- \(\langle 8\rangle\) in \(\mathbb{Q}^*\)
5. Explicitly identify the elements of the following subgroups of the given groups. You may use set-builder notation if the subgroup is infinite, or a conventional name for the subgroup if we have one.
- \(\langle 3\rangle\) in \(\mathbb{Z}\)
- \(\langle i\rangle\) in \(C^*\)
- \(\langle A\rangle\text{,}\) for \(A=\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right]\in \mathbb{M}_2(\mathbb{R})\)
- \(\langle (2,3)\rangle\) in \(\mathbb{Z}_4\times \mathbb{Z}_5\)
- \(\langle B\rangle\text{,}\) for \(B=\left[ \begin{array}{cc} 1 & 1\\ 0 & 1 \end{array} \right]\in GL(2,\mathbb{R})\)
6. Draw subgroup lattices for the following groups
- \(\mathbb{Z}_{6}\)
- \(\mathbb{Z}_{13}\)
- \(\mathbb{Z}_{18}\)
7. Let \(G\) be a group with no nontrivial proper subgroups. Prove that \(G\) is cyclic.