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5.3: Exercises

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    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 \(e\text{.}\)

    1. If \(G\) is infinite and cyclic, then \(G\) must have infinitely many generators.
    2. There may be two distinct elements \(a\) and \(b\) of a group \(G\) with \(\langle a\rangle =\langle b\rangle\text{.}\)
    3. If \(a,b\in G\) and \(a\in \langle b\rangle\) then we must have \(b\in \langle a\rangle\text{.}\)
    4. If \(a\in G\) with \(a^4=e\text{,}\) then \(o(a)\) must equal \(4\text{.}\)
    5. If \(G\) is countable then \(G\) must be cyclic.

    2. Give examples of the following.

    1. An infinite noncyclic group \(G\) containing an infinite cyclic subgroup \(H\text{.}\)
    2. An infinite noncyclic group \(G\) containing a finite nontrivial cyclic subgroup \(H\text{.}\)
    3. A cyclic group \(G\) containing exactly \(20\) elements.
    4. A nontrivial cyclic group \(G\) whose elements are all matrices.
    5. 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.

    1. \(2\in \mathbb{Z}\)
    2. \(-i\in \mathbb{C}^*\)
    3. \(-I_2\in GL(2,\mathbb{R})\)
    4. \(-I_2\in \mathbb{M}_2(\mathbb{R})\)
    5. \((6,8)\in \mathbb{Z}_{10}\times \mathbb{Z}_{10}\)

    4. For each of the following, if the group is cyclic, list all of its generators. If the group is not cyclic, write NC.

    1. \(5\mathbb{Z}\)
    2. \(\mathbb{Z}_{18}\)
    3. \(\mathbb{R}\)
    4. \(\langle \pi\rangle\) in \(\mathbb{R}\)
    5. \(\mathbb{Z}_2^2\)
    6. \(\langle 8\rangle\) 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.

    1. \(\langle 3\rangle\) in \(\mathbb{Z}\)
    2. \(\langle i\rangle\) in \(C^*\)
    3. \(\langle A\rangle\text{,}\) for \(A=\left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right]\in \mathbb{M}_2(\mathbb{R})\)
    4. \(\langle (2,3)\rangle\) in \(\mathbb{Z}_4\times \mathbb{Z}_5\)
    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

    1. \(\mathbb{Z}_{6}\)
    2. \(\mathbb{Z}_{13}\)
    3. \(\mathbb{Z}_{18}\)

    7. Let \(G\) be a group with no nontrivial proper subgroups. Prove that \(G\) is cyclic.

    This page titled 5.3: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jessica K. Sklar via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.