For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ is the trivial knot (bounding a disk) in a lens space $Y=L(p,q)$, then $\pi_1(Y-L)\cong \mathbb{Z}*\mathbb{Z}/p$. By definition, the link group is $(\mathbb{Z}*\mathbb{Z}/p)/[A]$, where $[A]$ is the commutator subgroup of ${\rm ker}(\pi_1(Y-L)\to \pi_1(Y)\cong \mathbb{Z}/p)$. Is there any explicit way to describe this link group? Especially, do we have $(\mathbb{Z}*\mathbb{Z}/p)/[A]\cong \mathbb{Z}*\mathbb{Z}/p$?
My question comes from the argument in https://en.wikipedia.org/wiki/Link_group, where it is said that the link group for trivial links in $S^3$ is isomorphic to a free group. However, in this question link group of the trivial $n$ component link, it seems that this argument is wrong since different articles use different definitions of "link group". Can we calculate any example of the link group in Milnor's sense explicitly?
