G1 - normal subgroups

normal subgroup

a subgroup $H$ of a group $G$ is called a normal subgroup of $G$ if $a H = H a \forall a \in G$ , denoted as $H ⊲ G$
alternatively, if $H$ is closed with respect to conjugates, ie. $x a x^{- 1} \in H \forall a \in H, x \in G$ , then $H ⊲ G$

note: $a H = H a ⟹ ⧸ a h = h a$ , but rather there is another element in $H$ such that $h a = a h^{'}$ , and there is another element in $H$ such that $a h = h^{″} a$ , where it may be possible that $h = h^{'}$ or $h = h^{″}$ , but not necessary
consider a group $G$ , generated by two elements $a$ and $b$ of order $3$ and $2$ respectively
if $a$ and $b$ commute, because it is a closed set ${e, a, a^{2}, b, a b, a b^{2}}$ , it is $G$ itself
else, all non-identity elements can be written as products of $a$ and $b$ with never more than two consecutive $a$ 's or two consecutive $b$ 's
some of them may be equal, and the group may be finite or infinite
if $H = {e, a, a^{2}}$ or $K = {e, b}$ is normal in $G$ , then $G = H K$
hence, normality here is as good as commutativity
eg: $N = {e, (123), (132)} ⊲ S_{3}$
$(23) N = {(23), (13), (12)}$ and $N (23) = {(23), (12), (13)}$ , so the left and the right cosets are equal
taking $x = (23) = x^{- 1} \in S_{3} :$ $(23) (123) (23) = (132) \in N$
$(23) (123) = (13)$ but $(123) (23) = (12)$ , so $a h \neq h a$
eg: a non example is $H = {e, (12)} < S_{3}$ because $(23) (12) (23) = (13) \notin H$