Group/Examples

Examples of Groups

Example: $\dfrac {x + y} {1 + x y}$

Let $G = \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.

Let $\circ: G \times G \to \R$ be the binary operation defined as:

$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$

The algebraic structure $\struct {G, \circ}$ is a group.

Example: $x + y + 2$ over $\R$

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

$\forall x, y \in \R: x \circ y := x + y + 2$

Then $\struct {\R, \circ}$ is a group whose identity is $-2$.

Example: $x + y + x y$ over $\R \setminus \set {-1}$

Let $\circ: \R \times \R$ be the operation defined on the real numbers $\R$ as:

$\forall x, y \in \R: x \circ y := x + y + x y$

Let:

$\R' := \R \setminus \set {-1}$

that is, the set of real numbers without $-1$.

Then $\struct {\R', \circ}$ is a group whose identity is $0$.

Example: $x^{-1} = 1 - x$

Let $S = \set {x \in \R: 0 < x < 1}$.

Then an operation $\circ$ can be found such that $\struct {S, \circ}$ is a group such that the inverse of $x \in S$ is $1 - x$.

Example: Group of Linear Functions

Let $G$ be the set of all real functions $\theta_{a, b}: \R \to \R$ defined as:

$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$

where $a, b \in \R$ such that $a \ne 0$.

The algebraic structure $\struct {G, \circ}$, where $\circ$ denotes composition of mappings, is a group.

$\struct {G, \circ}$ is specifically non-abelian.

Example: Operation Induced by Self-Inverse and Cancellable Elements

Let $S$ be a set with an operation which assigns to each $\tuple {a, b} \in S \times S$ an element $a \ast b \in S$ such that:

$(1): \quad \exists e \in S: a \ast b = e \iff a = b$

$(2): \quad \forall a, b, c \in S: \paren {a \ast c} \ast \paren {b \ast c} = a \ast b$

Then $\struct {S, \circ}$ is a group, where $\circ$ is defined as $a \circ b = a \ast \paren {e \ast b}$.

Example: $\tuple {a c, a d + b}$ on $\R_{\ge 0} \times \R$

Let $S = \R_{\ge 0} \times \R$ denote the Cartesian product of $\R_{\ge 0}$ and $\R$.

Let $\oplus: S \to S$ be the operation on $S$ defined as:

$\forall \tuple {a, b}, \tuple {c, d} \in S: \tuple {a, b} \oplus \tuple {c, d} := \tuple {a c, a d + b}$

Then the algebraic structure $\struct {S, \oplus}$ is a group.