Hall universal group as a direct limit of algebraic groups (Q1359015)

The Hall universal group \(U\), discovered by Philip Hall some forty years ago, is the unique countable locally finite group that contains copies of every finite group such that any two copies of the same finite group in \(U\) are conjugate. \(U\) is usually constructed as a direct limit of symmetric groups. The authors discuss ways in which \(U\) is a direct limit of locally finite (possibly finite) simple groups. Let \(F\) be a locally finite (possibly finite) field and \(G_n\) a classical simple group over \(F\) of rank \(n\). The authors prove that any infinite subsequence of \(\{G_n\}\) contains a subsequence whose direct limit is \(U\). (The embeddings \(G_r\to G_s\) here are not the natural embeddings.) The authors also show that \(U\) is not a direct limit of finitary nonlinear simple locally finite groups. In particular \(U\) is not a direct limit of infinite finitary alternating groups.