On recognizing groups by the bottom layer

Abstract

The article discusses the possibility of recognizing a group by the bottom layer, that is, by the set of its elements of prime orders. The paper gives examples of groups recognizable by the bottom layer, almost recognizable by the bottom layer, and unrecognizable by the bottom layer. Results are obtained for recognizing a group by the bottom layer in the class of infinite groups under some additional restrictions. The notion of recognizability of a group by the bottom layer was introduced by analogy with the recognizability of a group by its spectrum (the set of orders of its elements). It is proved that all finite simple nonAbelian groups are recognizable by spectrum and bottom layer simultaneously in the class of finite simple non-Abelian groups.