I'm all right in the proof till the point in which he says: the group $\langle a^2\rangle$ can contain no element of order $4$.
First: what does "CAN CONTAIN NO ELEMENT etc" mean exactly? English is not my mother tongue, however I think, literally, it seems that it means "there is the possibility that $\langle a^2\rangle$ doesen't contain any element of order $4$", or equivalently "$\langle a^2\rangle$ doesen't contain necessarely elements of order $4$". But this does not convince me. To me it would be more reasonable if this mean "$\langle a^2\rangle$ doesn't contain any element of order $4$".
Second: suppose what I meant is right ($\langle a^2\rangle$ doesn't contain any element of order $4$), let's understand why. If by contradiction $\langle a^2\rangle$ contain some such element, say $a^{2j}$, then it would be inverted by $a$, i.e. $(a^{2j})^a=a^{-2j}$, i.e. $a^{2j}=a^{-2j}$, i.e. $a^{4j}=1$. But from this I can't reach any contradiction. Can someone help me please? Thanks a lot!
You are right. The sentence means
In answer to your second question: If $a^{2j}$ has order 4, then $a^{4j} = (a^{2j})^2 \neq 1$.