On a quasivariety generated by a finite p-group

Abstract

Introduction. Let qG be the quasivariety generated by a group G and Lq(qG) the lattice of quasivarieties contained in qG. For a finite group G a method of construction of maximal proper quasivarieties in the lattice Lq(qG) has been found in [1]. The question arises whether it is true that each of these maximal quasivarieties is generated by some finite group. In this note an example of a finite group D is constructed such that the only maximal quasivariety in the lattice Lq(qD) is not generated by a finite group. We denote by Lp the set of quasivarieties each of which is generated by a finite p-group (p being a fixed prime). It is also proved in this article that L 2 is not a sublattice of the lattice of quasivarieties of groups. This provides a solution to Budkin's problem from the Kourovska Notebook (Problem 10.9 in [3]) for p = 2. 1. Preliminaries. Let 9~ be a quasivariety of groups and 97(G) the intersection of all nontrivial normal subgroups N of the group G such that G/N E {R. Definition. A nontrivial group G E 9~ is said to be subdirectly N-irreducible if {R(G) ;e (1). We denote by g the image of the element g E G under the natural homomorphism G --G/N. Fixing a prime p, we will use the following notation below: