= Torsion-free finite extension of a lattice
{title2=$1\to\mathbb Z^n\to\Gamma\to Q\to1$}
For a specified <finite group> action on $A=\mathbb Z^n$, extensions are classified by $e\in H^2(Q,A)$. For an element $q$ of prime order $p$, a lift $g$ satisfies $g^p=a\in A^q$; changing the lift changes $a$ by $N_qv=(1+q+\cdots+q^{p-1})v$. Thus the extension is torsion-free precisely when every restriction to a prime-order cyclic subgroup is nonzero in $A^q/N_qA$. Any finite-order element has a prime-order power, so this tests all torsion. An arbitrary finite quotient must not be confused with faithful holonomy on a maximal translation lattice.
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