I agree with you that this is a little delicate. I suppose it is more of a belief that $f$ can be deformed appropriately while maintaining $a_1=0$. If I think of a simple argument I will relate it.

Editted solution to reflect this.

You're absolutely right. I guess what I want is the following: suppose that $g$ moves clockwise $x$ times as fast as it moves anticlockwise. Then it spends $1/(x+1)$ of its time clockwise and $x/(x+1)$ of its time anticlockwise. Its speed clockwise is therefore $1+x$, anticlockwise $1+1/x$. I want then that $1+x−1=1+1/x+1$, i.e., $x^2−2x−1=0$.

I'm visualizing this backwards to the way it is proposed. Namely, if you don't generate half the group, take your set to be some $n+1$ of the elements you didn't reach.

A proof that the density of primes congruent to $m$ (modulo some $N$ presumably) is $1/\phi(m)$ is unlikely to be found anywhere.

Do you mean $np-q$? Because $p/q-1/n = (np-q)/nq$.

Why has this question been closed?

That's true, good points. His intention seems to ask whether an isomorphism $G\cong Z(G)\times G/Z(G)$ always holds. (On the other hand, I guess clearly doesn't hold, as it would imply that $Z(G) \cong Z(G) \times Z(G/Z(G))$ always holds.

Yes. I think we both posted at about the same time.