Group Homomorphism/Examples

Examples of Group Homomorphisms

Square Function

Let $\struct {\R_{\ne 0}, \times}$ be the group formed from the non-zero real numbers under multiplication.

Let $\struct {\R_{>0}, \times}$ be the group formed from the (strictly) positive real numbers under multiplication.

Let $f: \R_{\ne 0} \to \R_{>0}$ be the mapping defined as:

$\forall x \in \R_{\ge 0}: \map f x = x^2$

Then $f$ is a group homomorphism.

Mapping from Dihedral Group $D_3$ to Parity Group

Let $D_3$ denote the symmetry group of the equilateral triangle:

\(\ds e\)

\(:\)

\(\ds \paren A \paren B \paren C\)

Identity mapping

\(\ds p\)

\(:\)

\(\ds \paren {ABC}\)

Rotation of $120 \degrees$ anticlockwise about center

\(\ds q\)

\(:\)

\(\ds \paren {ACB}\)

Rotation of $120 \degrees$ clockwise about center

\(\ds r\)

\(:\)

\(\ds \paren {BC}\)

Reflection in line $r$

\(\ds s\)

\(:\)

\(\ds \paren {AC}\)

Reflection in line $s$

\(\ds t\)

\(:\)

\(\ds \paren {AB}\)

Reflection in line $t$

Let $G$ denote the parity group, defined as:

$\struct {\set {1, -1}, \times}$

where $\times$ denotes conventional multiplication.

Let $\theta: D_3 \to G$ be the mapping defined as:

$\forall x \in D_3: \map \theta x = \begin{cases} 1 & : \text{$x$ is a rotation} \\ -1 & : \text{$x$ is a reflection} \end{cases}$

Then $\theta$ is a (group) homomorphism, where:

\(\ds \map \ker \theta\)

\(=\)

\(\ds \set {e, p, q}\)

\(\ds \Img \theta\)

\(=\)

\(\ds G\)

Pointwise Addition on Continuous Real Functions on Closed Unit Interval

Let $J \subseteq \R$ denote the closed unit interval $\closedint 0 1$.

Let $\map {\mathscr C} J$ denote the set of all continuous real functions from $J$ to $\R$.

Let $G = \struct {\map {\mathscr C} J, +}$ be the group formed on $\map {\mathscr C} J$ by pointwise addition.

Let $\struct {\R, +}$ denote the additive group of real numbers.

From Pointwise Addition on Continuous Real Functions on Closed Unit Interval forms Group we have that $G$ is indeed a group.

Direct Sum

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be a homomorphism from $\struct {\map {\mathscr C} J, +}$ to $\struct {\R, +}$.

Let $\phi$ satisfy the condition:

$\forall c \in \R: \map \phi {f_c} = c$

where $f_c$ is the constant mapping on $\R$ defined as:

$\forall x \in \R: \map {f_c} x = c$

Then $\struct {\map {\mathscr C} J, +}$ is the internal group direct product of $\map \ker \phi$ and the subgroup of constant mappings on $\R$.

Example $1$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \map f 1$

Then $\phi$ is a homomorphism.

Example $2$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \size {\map f 0}$

where $\size {\, \cdot \,}$ denotes the absolute value function.

Then $\phi$ is not a homomorphism.

Example $3$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \map f x \rd x$

Then $\phi$ is a homomorphism.

Example $4$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \dfrac \pi 3 \int_0^1 \map f x \cos \dfrac {\pi x} 6 \rd x$

Then $\phi$ is a homomorphism.

Example $5$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \map \cos {\dfrac {\pi \map f x} 6} \rd x$

Then $\phi$ is not a homomorphism.

Example $6$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \map f {\cos \dfrac {\pi x} 6} \rd x$

Then $\phi$ is a homomorphism.

Example $7$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \int_0^1 \map f x \map f y \rd y \rd x$

Then $\phi$ is not a homomorphism.

Example $8$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds \int_0^1 \int_0^1 \map f {x y} \rd y \rd x$

Then $\phi$ is a homomorphism.

Example $9$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = \ds 2 \int_0^1 \int_0^x \map f y \rd y \rd x$

Then $\phi$ is a homomorphism.

Example $10$

Let $\phi: \struct {\map {\mathscr C} J, +} \to \struct {\R, +}$ be the mapping defined as:

$\forall f \in \map {\mathscr C} J: \map \phi f = -\map f 0 + \ds \int_{-2}^0 \map f {e^x} \rd x$

Then $\phi$ is a homomorphism.