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 ?

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    $\begingroup$ 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. $\endgroup$ – runway44 Oct 15 '20 at 5:22
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    $\begingroup$ Which book does you mean? Abstract Algebra or Topics in Algebra? $\endgroup$ – Alan Wang Oct 15 '20 at 5:37
  • $\begingroup$ 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 $\bar{e}$ ? $\endgroup$ – latus_rectum Oct 15 '20 at 5:39
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    $\begingroup$ @AlanWang Topics in Algebra, 2nd Edition $\endgroup$ – latus_rectum Oct 15 '20 at 5:41

I think your confusion arises from the wording "into".

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 $G$.

  • $\begingroup$ Got it. Also while defining isomorphism, Herstein states any homomorphism which is one-to-one is an isomorphism (whether it is onto or into). While in other text, it is stated that bijectivity is a must. Which of the two definitions should we go with ? $\endgroup$ – latus_rectum Oct 15 '20 at 6:09
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    $\begingroup$ @latus_rectum Bijectivity is a must. An isomorphism must be surjective (or onto). $\endgroup$ – Alan Wang Oct 15 '20 at 6:11

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