Show that it is false that every factor group of a nonabelian group is nonabelian.
I found this question while studying Fraleigh's algebra book, section 14, exercise 23, a true or false exercise. What is wrong with my thought:
There exist $g_1$ and $g_2$ of a nonabelian group G such that $g_1Hg_2H=g_1g_2H\neq g_2g_1H=g_2Hg_1H$. So there exist $g_1Hg_2H\neq g_2Hg_1H$.