$\textbf{1.1.} \enspace \triangleright$ Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category. $[\S2.1]$

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$\textbf{1.2.} \enspace \triangleright$ Consider the 'set of numbers' listen in $\S1.1$, and decide which are made into groups by conventional operations such as $+$ and $\cdot$. Even if the answer is negative (for example, $(\R,\cdot)$ is not a group), see if variations on the definition of these sets lead to groups (for example, ($\R^*,\cdot$) is a group; c.f. $\S1.4$). $[\S1.2]$

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$\textbf{1.12.} \enspace \triangleright$ In the group of invertible $2 \times 2$ matrices, consider

$$g= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad h= \begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix}.$$

Verify that $|g| = 4$, $|h| = 3$, and $|gh| = \infty. \enspace [\S1.6]$

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$\textbf{1.13.} \enspace \triangleright$ Give an example showing that $|gh|$ is not necessarily equal to $\text{lcm}(|g|, |h|)$, even if $g$ and $h$ commute. $[\S1.6, 1.14]$

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$\textbf{1.14.} \enspace \triangleright$ As a counterpoint to Exercise $1.13$, prove that if $g$ and $h$ commute and $\text{gcd}(|g|,|h|) = 1$, then $|gh| = |g| |h|$. (Hint: Let $N = |gh|$; then $g^N = (h^{-1})^N$. What can you say about this element?) $[\S1.6, 1.15, \S\text{IV}.2.5]$

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