# Group homomorphism

#### lemonthree

##### Member
Consider the group
.

"The map
defined by
for all
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

##### Well-known member
MHB Math Scholar
Hi lemonthree ,

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