MHB Group Homomorphism: True or False?

lemonthree · Sep 16, 2020

Consider the group

$(\mathbb{R}^*,\times)$

.

"The map

$\psi:\mathbb{R}^*\to\mathbb{R}^*$

defined by

$\psi(x)=|x|$

for all

$x\in \mathbb{R}^*$

is a group homomorphism."

Is this true or false? I'm guessing it's true because
φ (j) = | j |, which means
φ (j * k) = | j * k |
=| j | * | k |
= φ ( j ) * φ ( k ).

castor28 · Sep 16, 2020

Hi lemonthree,

That is true, but you should also check that

$\phi(x^{-1}) = \phi(x)^{-1}$
$\phi(1) = 1$

MHB Group Homomorphism: True or False?

Thread 'When are Markov Matrices also Martingales?'

Thread 'Derivation of equations of stress tensor transformation'

Similar threads

Hot Threads

I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?

I Showing ##k[x_1,\ldots,x_n]/\mathfrak{a}## is finite dimensional

I How do we distinguish two different notations for cokernel and coimage?

I Localising a non integral domain at a prime

I ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?

Recent Insights

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Insights What Exactly is Dirac’s Delta Function? - Insight

Insights Relativator (Circular Slide-Rule): Simulated with Desmos - Insight

Insights Fixing Things Which Can Go Wrong With Complex Numbers