Locally finite finitary skew linear groups (Q2766410)

Let \(D\) be a division ring and \(V\) a left vector space over \(D\). The set \(\text{FGL}(V)\) of all elements \(g\) in \(\text{GL}(V)\) satisfying \(\dim_DV(g-1)<\infty\) is a subgroup of \(\text{GL}(V)\). Any subgroup of \(\text{FGL}(V)\) is called a finitary skew linear group. The main result of this paper is as follows. Let \(G\) be a locally finite, primitive subgroup of \(\text{FGL}(V)\), where \(\dim_DV\) is infinite. Then the derived subgroup \(G'\) of \(G\) is the unique minimal normal subgroup of \(G\), \(G'\) is an infinite simple group lying in every non-trivial normal subgroup of \(G\), \(C_G(G')=\langle 1\rangle\), and \(G'\leq G\leq\Aut G'\). Moreover, if \(\text{char }D=0\), then \(G\) is isomorphic to either \(\text{FSym}(\Omega)\) or \(\text{Alt}(\Omega)\) for some infinite set \(\Omega\), where \(\text{FSym}(\Omega)\) is the finitary symmetric group on the set \(\Omega\) and \(\text{Alt}(\Omega)\) is the alternating group on \(\Omega\). In both cases, \(G'\) is isomorphic to \(\text{Alt}(\Omega)\). From this theorem, the author deduces several important corollaries.