# Thread: Normal endomorphism of a group

1. ## Normal endomorphism of a group

-----------------------------------------------------------------
(1) An endomorphism f of a group G is called normal endomorphism if $\displaystyle af(b)a^{-1} = f(aba^{-1})$ for all $\displaystyle a,b \in G$.
(2) f + g denotes the function G->G given by $\displaystyle a \mapsto f(a)g(a)$.
--------------------------------------------------------------------
[Hungerford, p88, Q3.8]

Let f and g be normal endomorphisms of a group G. Prove that if f + g is an endomorphism, then it is normal.

2. Originally Posted by aliceinwonderland
-----------------------------------------------------------------
(1) An endomorphism f of a group G is called normal endomorphism if $\displaystyle af(b)a^{-1} = f(aba^{-1})$ for all $\displaystyle a,b \in G$.
(2) f + g denotes the function G->G given by $\displaystyle a \mapsto f(a)g(a)$.
--------------------------------------------------------------------
[Hungerford, p88, Q3.8]

Let f and g be normal endomorphisms of a group G. Prove that if f + g is an endomorphism, then it is normal.
This is just a straightforward computation. Let $\displaystyle h=f+g$ be the function $\displaystyle a\mapsto f(a)g(a)$.
If $\displaystyle h$ is an endomorphism then $\displaystyle h(aba^{-1}) = f(aba^{-1})g(aba^{-1}) = (af(b)a^{-1} )(ag(b)a^{-1})$ so $\displaystyle h(aba^{-1}) = af(b)g(b)a^{-1} = ah(b)a^{-1}$.
This shows that $\displaystyle h$ is a normal endomorphism.

3. [Edited] I removed another question, coz it turned out trivial.