A test for a local formation of finite groups to be a formation of soluble groups with the Shemetkov property

Abstract

L.A. Shemetkov posed a Problem 9.74 in Kourovka Notebook to find all local formations F of finite groups such that every finite minimal non-F-group is either a Schmidt group or a group of prime order. All known solutions to this problem are obtained under the assumption that every minimal non-F-group is soluble. Using the above mentioned solutions we present a polynomial in n time check for a local formation F with bounded π(F) to be a formation of soluble groups with the Shemtkov property where n = max π(F).