##### 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.

\(2\in \Z\)

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

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

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

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

##### 4

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

\(5\Z\)

\(\Z_{18}\)

\(\R\)

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

\(\Z_2^2\)

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

##### 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 \(\Z\)

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

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

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

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

##### 6

Draw subgroup lattices for the following groups.

\(\Z_6\)

\(\Z_{13}\)

\(\Z_{18}\)

##### 7

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