In this paper, the authors determine the group \(T(G)\) of endotrivial \(kG\)-modules for an algebraically closed field \(k\) of characteristic 2 and a group \(G\) whose Sylow 2-subgroups are either semi-dihedral or generalized quaternion.
Recall that a \(kG\)-module \(M\) is said to be endotrivial if its endomorphism algebra \(\mathrm{End}_k(M)\cong M^*\otimes_kM\) is, as a \(kG\)-module, stably isomorphic to the trivial simple \(kG\)-module \(k\). If \(M\) is an endotrivial \(kG\)-module, then there exists a unique (up to isomorphism) indecomposable endotrivial \(kG\)-module \(M_0\) such that \(M\) is stably isomorphic to \(M_0\). Two endotrivial \(kG\)-modules \(M\) and \(N\) are said to be equivalent if \(M_0\cong N_0\). The set \(T(G)\) is defined to be the set of all equivalence classes of endotrivial \(kG\)-modules, which becomes an Abelian group using the tensor product \(\otimes_k\).
Let \(P\) be a Sylow 2-subgroup of \(G\), which is by assumption either semi-dihedral or generalized quaternion. In both cases, the authors prove that the restriction map \(T(G)\to T(P)\) is surjective. They then determine the group \(T(G)\) in detail. In the semi-dihedral case, the key ingredient in the proofs is the theory of almost split sequences. In particular, they use that there exists an almost split sequence whose middle term is stably isomorphic to the heart \(\mathrm{rad}(Q_k)/\mathrm{soc}(Q_k)\) of the projective cover \(Q_k\) of the trivial module \(k\). In the quaternion case, they first analyze the case when the unique involution \(z\) of \(P\) is central in \(G\). For the general case, they use that the centralizer of \(z\) is a strongly 2-embedded subgroup of \(G\). They moreover show that there are always torsion endotrivial modules which are uniserial.