Group Homomorphism/Examples/Pointwise Addition on Continuous Real Functions on Closed Unit Interval/Example 5

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Example of Group Homomorphism

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.


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.


Proof

Let $f, g \in \map {\mathscr C} J$ be arbitrary.


As both $f$ and $g$ are continuous real functions on $J$, both $\map \cos {\dfrac {\pi \map f x} 6}$ and $\map \cos {\dfrac {\pi \map g x} 6}$ are integrable on $\closedint 0 1$.


We have:

\(\ds \map \phi {f + g}\) \(=\) \(\ds \int_0^1 \map \cos {\dfrac {\pi \map {\paren {f + g} } x} 6} \rd x\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \int_0^1 \map \cos {\dfrac {\pi \paren {\map f x + \map g x} } 6} \rd x\) Definition of Pointwise Addition of Real-Valued Functions
\(\ds \) \(=\) \(\ds \int_0^1 \paren {\map \cos {\dfrac {\pi \map f x} 6} \map \cos {\dfrac {\pi \map g x} 6} - \map \sin {\dfrac {\pi \map f x} 6} \map \sin {\dfrac {\pi \map g x} 6} } \rd x\) Cosine of Sum

However:

\(\ds \map \phi f + \map \phi g\) \(=\) \(\ds \int_0^1 \map \cos {\dfrac {\pi \map f x} 6} \rd x + \int_0^1 \map \cos {\dfrac {\pi \map g x} 6} \rd x\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \int_0^1 \paren {\map \cos {\dfrac {\pi \map f x} 6} + \map \cos {\dfrac {\pi \map g x} 6} } \rd x\) Linear Combination of Definite Integrals

Thus:

$\map \phi {f + g} \ne \map \phi f + \map \phi g$

and so $\phi$ is not a homomorphism by definition.

$\blacksquare$


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 $13.18 \ \text {(e)}$
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