Examples Groups

1.

$(\mathbb{Z}, +)$, the set of integers with addition.\\ The neutral element is $e=0$,\\ The inverse element for $n \in \mathbb{Z}$ is $-n \in \mathbb{Z}$.\\ This also holds for $(\mathbb{Q}, +)$ and $(\mathbb{R}, +)$.

2.

$(\mathbb{R}^\times, \cdot)$, where $\mathbb{R}^\times = \mathbb{R} \setminus \{0\}$.\\ The neutral element is $e=1$,\\ The inverse element for $a \in \mathbb{R}$ is $\frac{1}{a} \in \mathbb{R}$.\\ This also holds for $(\mathbb{Q}^\times, \cdot)$ and $(\mathbb{R}_{>0}, \cdot)$.

3.

$(\{\pm 1\}, \cdot)$ is a group with 2 elements. The group table is \begin{align*} \begin{array}{c|cc} \cdot & +1 & -1 \\ \hline +1 & +1 & -1 \\ -1 & -1 & +1 \\ \end{array} \end{align*}

4.

Here we consider, for a natural number $n$, the set $\mathbb{Z} / n \mathbb{Z}$ of equivalence classes of $\mathbb{Z}$ with respect to the equivalence relation congruence modulo $n$, from Chapter 1.7.3.\\ We define an internal operation on $\mathbb{Z} / n \mathbb{Z}$ by \begin{align*} \left[ a \right] _n + \left[ b \right]_n := \left[ a + b \right] _n. \end{align*} Here, $a, b \in \mathbb{Z}$ are representatives of the corresponding equivalence classes. We also call $[a + b]_n$ the \textit{sum of the equivalence classes} $[a]_n$ and $[b]_n$. The definition of this sum is independent of the choice of representatives and thus well-defined. For if $a' \in [a]_n$ and $b' \in [b]_n$ are other representatives, it holds that $$ [a' + b']_n = [a + b]_n. $$ The set $\mathbb{Z}/n\mathbb{Z}$ with this sum of equivalence classes forms an abelian group.

5.

Let $M$ be a set. Define $$ S(M) = \{ f : M \to M \mid f \text{ is bijective} \}, $$ the set of bijective mappings from $M$ to $M$, and $$ \circ : S(M) \times S(M) \to S(M), \quad (f, g) \mapsto f \circ g, $$ the composition of mappings. Thus, $\left( S(M), \circ \right)$ is a group.\\ The neutral element is the identity mapping, $\text{id}_M : M \to M, \, m \mapsto m$.\\ The inverse element for $f$ is $f^{-1}$, the inverse mapping to $f$.