I am slightly confused about the definition of normal subgroups. The book gives the definition that a subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a$ in $G$. Then it explains that one can switch the order of a product of an element $a$ from the group and the element $h$ from $H$, but one must fudge a bit on the element $h$, by using some $h'$ instead of $h$. That is, there is an element $h'$ in $H$ such that $ah=h'a$. then it says it is possible that we can have $h'=h$, but we cannot assume this
Why do we need to change the element $h$ to $h'$? I do not understand it. Also, why cannot we say $h=h'$?
Any help would be appreciated!
Thanks in advance!
Another way of phrasing $aH = Ha$ is to say that $a^{-1}Ha = H$. In other words, consider the map $\varphi_a \colon G \to G$ which is conjugation by $a$:
$$ \varphi_a(g) = a^{-1}ga $$
These types of maps (given by conjugation) are called "inner automorphisms". Being a normal subgroup means $H$ is invariant under all inner automorphisms. That is, $\varphi(H) = H$. The crucial thing to point out here is that this DOES NOT mean that $\varphi(h) = h$ for every $h$ in $H$. The notation $\varphi(H) = H$ simply means that $\varphi(h) \in H$ for every $h \in H$.
On a side note, if it were true that $ah = ha$ for every $h \in H$ and every $a \in G$, this would mean that $H$ is in the center of $G$ (elements of $H$ commute with everything), which does not have to be true in general for a normal subgroup. For a concrete example, consider $\mathrm{Alt}(5) \unlhd \mathrm{Sym}(5)$. The alternating group is a normal subgroup of the symmetric group, but the alternating group itself is not even abelian, so it certainly can't commute with all permutations.