First Tor with a cyclic group

Exercise 6.5.8. The derived \(\infty \)-category \(\D (\Z )\) carries a symmetric monoidal structure \(\otimes ^{\bL }\) with unit \(\Z \); see Section 6.6. For an abelian group \(M\), use the short exact sequence \[ 0 \to \Z \xrightarrow {\cdot n} \Z \to \Z /n\Z \to 0 \] to show that \(\Tor ^{\otimes ^{\bL }}_1(M, \Z /n\Z )\) is the subgroup of \(M\) consisting of the \(n\)-torsion elements, i.e., those \(x \in M\) satisfying \(nx = 0\).