(related to Chapter: Groups (Overview))

The following are examples of groups and the reader is invited to verify the defining properties of a group for each of the examples.

- The set $(\mathbb Z, + )$ of integers, together with addition is a group.
- The set of $(\mathbb Q, + )$ of rational numbers under addition forms a group.
- The set of non-zero $(\mathbb Q^*,\cdot)$ of rational numbers under multiplication forms a group.
- The set of $(\mathbb R,+ )$ of real numbers under addition forms a group.
- The set of non-zero $(\mathbb R^*,\cdot)$ of real numbers under multiplication forms a group.
- The set of $(\mathbb C,+ )$ of complex numbers under addition forms a group.
- The set of non-zero $(\mathbb C^*,\cdot)$ of complex numbers under multiplication forms a group.
- Let $(G,\ast)$ be a group and let $S$ be a non-empty set. The set $(M,\circ)$ of maps $f,g:S\to G$ is itself a group for each $x\in S$, with the binary operation $(f\circ g)(x):=f(x)\ast g(x)$ and the inverse $f^{-1}(x):=(f(x))^{-1}\in G$. The identity element is the map $e'\in M$ which maps the element $x$ to the identity element $e\in G$.
- Let $S$ be a non-empty set and let $G$ be the set of bijective mappings $f:S\to S$. Then $(G,\circ)$ is a group with the composition of mappings $(f\circ g)(x):=f(g(x))$ for all $x\in S$. The identity element is the identity map $id:S\to S$ and, since all mappings $f\in G$ are bijective, $G$ contains inverse mappings $f^{-1}\in G$ for all $f\in G$. This example is called a
*symmetric group.* - Let $V$ be a vector space over a field $F$. Let $GL(V)$ denote the set of invertible linear maps of $V$ onto itself. Then $(GL(V),\circ)$ is a group, where the binary operation $\circ$ denotes the composition of mappings.
- Let $V$ be a vector space over a field $F$. Let $GL(V,F)$ denote the set of invertible $n\times n$ matrices with components in $F$. Then $(GL(F),\circ)$ is a group, where the binary operation $\circ$ denotes the matrix multiplication. This group is called the
**general linear group**. - The direct product of groups is a group.

The following are examples which are not groups:

- The set $(\mathbb N, + )$ of natural numbers, together with addition is not a group since not all natural numbers have an inverse element. In fact, only $0\in \mathbb N$ is the only element with the inverse $-0\in\mathbb N.$
- The set $(\mathbb Z, \cdot )$ of integers together with multiplication is not a group, since not all elements $a\in \mathbb Z$ have an inverse, e.g. $2\in \mathbb Z$ but $\frac 12\not\in\mathbb Z.$
- The set $(\mathbb R,\ast)$ of real numbers, together with the mean value operation $a \ast b=\frac{a+b}2$ is not a group, since this operation is not associative.

**Lang, Serge**: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition