Proof of the Day: Homomorphisms and Abelian Groups

Let f:G \rightarrow H be a homomorphism of Abelian groups.  Show that f(-x)=-f(x) for each x\epsilon G.


Since f is a homomorphism, and since x\epsilon G, an Abelian group, then:

f(x) = f(-x)+0

= f(-x) + (f(x) - f(x))

= (f(-x) + f(x)) - f(x) (by associativity)

= f(-x+x)-f(x) (f is a homomorphism)

=  f(0)-f(x)

= -f(x)

The last line is true because f(0)=0 (but that’s another proof).