Nilpotent cofinitary groups (Q1903031)

Let \(D\) be a division ring, \(V\) a vector space over \(D\) of infinite dimension. Say that an element \(g \in \text{GL} (V)\) is cofinitary if \(\dim_D C_V (g)\) is finite. A subgroup \(G \leq \text{GL} (V)\) is called cofinitary if all its non-trivial elements are cofinitary. Note that the set of cofinitary elements of \(\text{GL} (V)\) does not form a subgroup, unlike the finitary case. A subgroup \(G\) of \(\text{GL} (V)\) is irreducible if \(V\) is irreducible as a \(D\)-\(G\)-bimodule and primitive if \(V\) does not properly decompose into a direct sum of \(D\)-submodules permuted by \(G\). The main result is Theorem 1.1. Let \(G\) be an irreducible subgroup of \(\text{GL} (V)\) where \(\dim_D V\) is infinite. Suppose that for some \(c \geq 0\) the subgroup \(\zeta_{c + 1} (G)\) is cofinitary and contains an element \(g \neq 1\) with non-zero fixed point in \(V\). If the torsion part of \(\zeta_1(G)\) is bounded, then \(G\) is imprimitive. 1.2 Corollary. Let \(G\) be a nilpotent irreducible cofinitary subgroup of \(\text{GL} (V)\) and suppose that the torsion part of \(\zeta_1 (G)\) is bounded. Then either (a) \(\dim_D V\) is finite; or (b) \(G\) is imprimitive: or (c) \(G\) acts fixed-point freely on \(V\).