Example of a Subgroup That Is Not Normal

Solution 1:

Note that in your proof, $xy = (a_1h_1)(b_1h_1^*)$. When you write that $xy = (a_1b_1)(h_1h_1^*)$ is assuming the group in question is abelian. But this assumption is unwarranted, because when commutativity does not hold, we cannot assume $xy = (a_1h_1)(b_1h_1^*)=(a_1b_1)(h_1h_1^*)$.

Indeed, if a group is abelian, then every one of its subgroups are normal, as you've shown to be the case, but this doesn't hold in nonabelian groups.

Consider, for example, the group $G = S_3$ and the subgroup $H\lt G$, $H = \{id, (12)\}$. This group is not normal in $S_3$. Consider other non-abelian groups, as well, to find many other counterexamples.