Diagonal limits of finite alternating groups: confined subgroups, ideals, and positive definite functions.
It is difficult to describe briefly the contents of this substantial paper without getting technical. It concerns certain locally finite simple groups \(G\); specifically those \(G\) that are non-finitary and are direct limits of finite alternating groups \(G_i=\text{Alt}(Y_i)\) for \(i\) in \(I\) such that each \(G_i\) has only trivial or natural orbits on the \(Y_j\) for \(j>i\). A non-trivial transitive permutation representation of \(G\) on a set \(Y\) is natural if every \(G_i\) has only trivial or natural orbits on \(Y\). A subgroup \(X\) of \(G\) is confined in \(G\) if there is a finite subgroup \(F\) of \(G\) such that every conjugate of \(X\) in \(G\) contains a non-trivial element of \(F\). The authors' first main result is the following. Let \(X\) be a proper subgroup of \(G\). If \(X\) is the intersection of finitely many point stabilizers of natural permutation representations of \(G\), then every overgroup of \(X\) in \(G\) is confined in \(G\). Conversely, if \(X\) is confined in \(G\) with respect to the finite subgroup \(F\), then \(X\) contains a simple normal subgroup \(N\), which is an intersection of such point stabilizers, and whose index in \(X\) is finite and bounded by a function of the order of \(F\). Let \(K\) be a field of characteristic zero. The above is related to the lattice of ideals in the group algebra \(KG\). The ideals in \(KG\) in fact form a descending chain of length \(\omega+1\). The authors show that if \(X\) is a confined subgroup of \(G\) with a simple normal subgroup of finite index that is the intersection of \(d\) pairwise distinct point stabilizers of natural permutation representations of \(G\), then the intersection of the \((X^g-1)KG\) as \(g\) runs over \(G\) is the \(d\)-th non-trivial ideal in this descending chain of ideals of \(KG\). The third strand of the paper considers the special case where \(K\) is chosen to be the field of complex numbers.
scientific article; zbMATH DE number 1993640