Infinite alternating groups as finitary linear transformation groups
Let V be a vector space over a field k. A finitary linear transformation of V is an invertible linear map g: \(V\to V\) such that V(g-1) is finite- dimensional, or equivalently, such that g acts as the identity on a subspace of finite codimension. The set of all such g is a group FGL(V) under composition of maps, the finitary general linear group on V. This interesting paper launches a program to find all infinite simple locally finite groups that can be embedded in some FGL(V). (Note that an easy extension of Schur's well known result proves that every periodic subgroup of FGL(V) is locally finite.) The case when V is finite dimensional was dealt with by a number of authors (see the paper for details), using the classification of finite simple groups. A natural example requiring an infinite dimensional space is the alternating group Alt(\(\Omega)\) on an infinite set \(\Omega\), acting on the space \(k\Omega\). The author's main theorem is that in characteristic 0, this is the only possibility: Let G be an infinite, simple, locally finite subgroup of FGL(V), where V is a vector space over a field of characteristic zero. Then V is isomorphic to an alternating group. It was suggested by \textit{A. Zalesskij} that such a result might be true [Dokl. Akad. Nauk BSSR 13, 485-488 (1969; Zbl 0223.20046)], though the author was evidently unaware of this at the time the paper was written. This beautiful result also depends on the classification of finite simple groups, as well as a substantial amount of information about the structure and properties of these groups. The possible finitary representations of Alt(\(\Omega)\) are also considered. Let V(k\(\Omega)\) denote the ``sum zero'' submodule of \(k\Omega\). Theorem 8.1 asserts that if V is a vector space over a field of characteristic zero, \(G\leq FGL(V)\) and \(G\cong Alt(\Omega)\) for an infinite set \(\Omega\), then [V,G] is a direct sum of a finite number of copies of V(k\(\Omega)\). Other interesting facts about Alt(\(\Omega)\) emerge. Theorem 2: Let G be a group in which each countable subset lies in a subgroup isomorphic to an alternating group. Then G is isomorphic to an alternating group. The link between infinite simple groups and finite simple groups is made through the notion of a sectional cover. A sectional cover for a locally finite group G is a set of pairs \(\{F_{\lambda},N_{\lambda}\}_{\lambda \in \Lambda}\) such that \(F_{\lambda}\) is a finite subgroup of G, \(N_{\lambda}\triangleleft F_{\lambda}\), and for each finite subgroup H of G, there exists \(\lambda\in \Lambda\) with \(H\leq F_{\lambda}\) and \(H\cap N_{\lambda}=1\). A well known result of Kegel assures us that if G is a simple locally finite group, then G has a sectional cover \(\{F_{\lambda},N_{\lambda}\}\) in which each \(F_{\lambda}/N_{\lambda}\) is simple. Here we have the following theorem: Theorem (5.3) Let H be a locally finite group contained in FGL(V), where V is a vector space over a field of arbitrary characteristic. Suppose that \(G=H/K\) is an infinite simple quotient of H, and that G has a sectional cover \(\Delta\) in which F/N is alternating for each (F,N)\(\in \Delta\). Then \(G\cong Alt(\Omega)\) for some \(\Omega\). An important notion throughout is that of abstract degree, which seems to arise naturally in this context. Let K be a subgroup of a finite group L. The abstract degree of K in L, written \(a\deg_ L(K)\), is the minimum value of \(\dim_ k[V,\phi (K)]\), where \(\phi\) denotes a faithful projective representation of L on the vector space V over the field k, and the minimum is taken over all choices of V, k and \(\phi\). A substantial part of the paper is taken up with discussing the behaviour of this function and its counterpart in the locally finite situation.
infinite, simple, locally finite subgroup of FGL(V)
scientific article; zbMATH DE number 4087913
scientific article