MHB Group Homomorphism: True or False?

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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 ).
 
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Hi lemonthree,

That is true, but you should also check that
  • $\phi(x^{-1}) = \phi(x)^{-1}$
  • $\phi(1) = 1$
 

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