Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval
Examples of Group Homomorphisms
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.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.18$