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

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

Group Homomorphism: True or False?

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