Torsion-free space groups, \(2\)-cohomology, and Scott modules (Q1911792)

Let \(C\) be a subgroup of prime order \(p\) of the finite group \(G\), and suppose that the induced module \(\text{ind}^G_C\mathbb{Z}_p=\bigoplus_i M_i\), where \(\mathbb{Z}_p\) denotes the ring of \(p\)-adic integers, regarded as a trivial \(\mathbb{Z}_pC\)-module, and each \(M_i\) is an indecomposable \(\mathbb{Z}_pG\)-module. Let \(M_1\) be the summand with the second cohomology \(H^2(G,M_1)\) non zero. Then the main result of the present paper is to show that \(M_1\) is the Scott module in \(\text{ind}^G_C\mathbb{Z}_p\) (i.e., \(M_1\) is the unique summand of \(\text{ind}^G_C\mathbb{Z}_p\) containing a trivial submodule) if and only if the normalizer of \(C\) in \(G\) is equal to the centralizer of \(C\) in \(G\). The above result was suggested by Plesken in private communication to the authors. The authors apply the above result to the study of torsion-free space groups in the case of the alternating group \(A_5\).