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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)



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.


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.


Find the orders of the following elements in the given groups.

  1. \(2\in \Z\)

  2. \(-i\in \C^*\)

  3. \(-I_2\in GL(2,\R)\)

  4. \(-I_2\in \M_2(\R)\)

  5. \((6,8)\in \Z_{10}\times \Z_{10}\)


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

  1. \(5\Z\)

  2. \(\Z_{18}\)

  3. \(\R\)

  4. \(\langle \pi\rangle\) in \(\R\)

  5. \(\Z_2^2\)

  6. \(\langle 8\rangle\) in \(\Q^*\)


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 \(\Z\)

  2. \(\langle i\rangle\) in \(C^*\)

  3. \(\langle A\rangle\text{,}\) for \(A=\left[ \begin{array}{cc} 1 \amp 0 \\ 0 \amp 0 \end{array} \right]\in \M_2(\R)\)

  4. \(\langle (2,3)\rangle\) in \(\Z_4\times \Z_5\)

  5. \(\langle B\rangle\text{,}\) for \(B=\left[ \begin{array}{cc} 1 \amp 1\\ 0 \amp 1 \end{array} \right]\in GL(2,\R)\)


Draw subgroup lattices for the following groups.

  1. \(\Z_6\)

  2. \(\Z_{13}\)

  3. \(\Z_{18}\)


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