Originally Posted by

**aliceinwonderland** Thanks for all replies.

I have one last paragraph in my text that I could not understand regarding a nonabelian group of order 12.

"Let a = cd (in the above quote); then <a> is normal in G and |G/<a>| = 2. Hence there is an element $\displaystyle b \in G$ such that $\displaystyle b \notin <a>, b \neq e, b^{2} \in <a>, bab^{-1} \in <a>$.

Since G is nonabelian and $\displaystyle |a|=6, bab^{-1} = a^{5} = a^{-1}$ is the only possibility.

....

There are six possibilities for $\displaystyle b^{2} \in <a>. b^{2}=a^{2}, b^{2}=a^{4} $ **lead to contradictions.** $\displaystyle b^{2}=a$ or $\displaystyle b^{2}=a^{5}$ imply |b|=12.

..........."

Why the $\displaystyle b^{2}=a^{2}, b^{2}=a^{4}$ lead to contradiction?