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## 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 $x\epsilon G$.

Proof:

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).

$\Box$