Elementary proof of a theorem of Blackburn

Abstract

For a positive integer n, a finite p-group G is called an ℳ_n-group, if all subgroups of index pⁿ of G are metacyclic, but there is at least one subgroup of index pⁿ⁻¹ that is not. A classical result in p-group theory is the classification of ℳ₁-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result.

Keywords

Finite p-group / ℳ_n-group / minimal non-abelian group / minimal non-metacyclic group

References

[1] Blackburn N. Generalizations of certain elementary theorems on p-groups. Proc London Math Soc, 1961, 11(3): 1-22.

[2] Burnside W. Theory of Groups of Finite Order, 1897, Cambridge: Cambridge University Press.

[3] Huppert B. Endliche Gruppen I, 1967, Berlin: Springer-Verlag.

[4] Rédei L. Das schiefe Product in der Gruppentheorie. Comment Math Helvet, 1947, 20: 225-267.