On subgroups of finite \(p\)-groups.
Let \(G\) be a finite nonabelian \(p\)-group, where \(p\) is a prime number. By a classical result of Blackburn if the group \(G\) is 2-generated with 2-generated maximal subgroups then it is essentially metacyclic. For \(p=2\) the authors classify groups \(G\) having all of their nonabelian maximal subgroups 2-generated; they turn out to be either metacyclic or minimal nonabelian. In the odd prime \(p\) case neither metacyclic nor minimal nonabelian 2-generated groups \(G\) having all of their nonabelian maximal subgroups 2-generated are studied; namely, in the presence of an Abelian maximal subgroup the factor \(G/\mho_1\) is described. The authors, among 20 research problems, pose the question of determining groups \(G\) with all of their minimal nonabelian subgroups isomorphic; as a first step in this direction they examine 2-groups \(G\) with all of their minimal nonabelian subgroups isomorphic to the group \(\langle a,b\mid a^8=1,\;b^2=1,\;a^b=a^5\rangle\) of order 16. Finally, the authors consider a nonmetacyclic 2-generator 2-group \(G\). It is already known that it should contain a maximal subgroup which is not 2-generated; several other properties are proved, such as that the number of 2-generated as well as metacyclic maximal subgroups is even and, moreover, such a group containing three maximal subgroups is described.
scientific article; zbMATH DE number 5600318
scientific article