Is the Kernel of a homomorphism normal only when the homomorphism is into ? Why?

In the Text Herstein's Algebra :

In the Homomorphism section, he defines Kernel as :

If $$\phi$$ is a homomorphism of G into G, the kernel of $$\phi$$, $$K_\phi$$, is defined by $$K_\phi$$ = {$$x \in G$$ | $$\phi(x)$$ = e, e = identity element of G}.

And then states this particular Lemma :

If $$\phi$$ is a homomorphism of G into $$\bar{G}$$ with kernel K, then K is a normal subgroup of G.

Here, why the mapping between G and $$\bar{G}$$ has to be into ? Will K be a normal subgroup if mapping is onto ?

• It seems like the word "into" is not referring to a property the homomorphism has, it's just saying what the codomain is. Kernels are always normal. – runway44 Oct 15 '20 at 5:22
• Which book does you mean? Abstract Algebra or Topics in Algebra? – Alan Wang Oct 15 '20 at 5:37
• When he talks about inverse images of $\bar{g}$ other than $\bar{e}$, then he defines $\phi$ as a homomorphism of G onto $\bar{G}$. Why this difference in defining Kernel (i.e. all inverse of $\bar{g}$ equal to $\bar{e}$) and when defining all inverse for $\bar{g}$ other than $\bar{e}$ ? – latus_rectum Oct 15 '20 at 5:39
• @AlanWang Topics in Algebra, 2nd Edition – latus_rectum Oct 15 '20 at 5:41

For a function $$f:A\rightarrow B$$, we normally say that $$f$$ is a function from $$A$$ to $$B$$. Although in this book the author sometimes said $$f$$ is a function (or mapping) from $$A$$ into $$B$$, they are basically the same thing. They mean that for every $$a\in A$$, there must be a unique element $$b\in B$$ such that $$f(a)=b$$.
Let $$G,\bar{G}$$ be two groups.
As long as $$f:G\rightarrow \bar{G}$$ is a homomorphism (that is, $$f:G\rightarrow \bar{G}$$ is a well-defined function such that $$f(xy)=f(x)f(y)$$ for all $$x,y\in G$$), the kernel of $$f$$ is always a normal subgroup of $$G$$.
So if furthermore the function $$f$$ is onto (or surjective), the kernel of $$f$$ is still normal in $$G$$.