= Solution
Every finite quotient of the abelian group $A_{\neg p}$ is abelian. Part iii excludes elements of prime order $q\ne p$ by the <Cauchy theorem for groups>, so any finite quotient is a finite abelian $p$-group. If its exponent divides $p^n$, the quotient map kills $p^nA_{\neg p}$ and therefore factors through
$$
A_{\neg p}/p^nA_{\neg p}\cong\mathbb Z/p^n\mathbb Z.
$$
Every quotient of a <cyclic group> is cyclic, so the finite quotient is isomorphic to $\mathbb Z/p^k\mathbb Z$ for some $k\leq n$. Conversely, reduction modulo $p^k$ gives a surjection $A_{\neg p}\to\mathbb Z/p^k\mathbb Z$. Thus these are exactly the nontrivial finite quotients, together with the trivial case $k=0$.
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