Glasnik Matematicki, Vol. 41, No.2 (2006), 239-258.

SHORT PROOFS OF SOME BASIC CHARACTERIZATION THEOREMS OF FINITE p-GROUP THEORY

Yakov Berkovich

Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
e-mail: berkov@math.haifa.ac.il


Abstract.   We offer short proofs of such basic results of finite p-group theory as theorems of Blackburn, Huppert, Ito-Ohara, Janko, Taussky. All proofs of those theorems are based on the following result: If G is a nonabelian metacyclic p-group and R is a proper G-invariant subgroup of G', then G/R is not metacyclic. In the second part we use Blackburn's theory of p-groups of maximal class. Here we prove that a p-group G is of maximal class if and only if Ω2*(G) = 〈 x G | o(x) = p2 is of maximal class. We also show that a noncyclic p-group G of exponent > p contains two distinct maximal cyclic subgroups A and B of orders > p such that |AB| = p, unless p = 2 and G is dihedral.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Finite p-groups, metacyclic p-groups, minimal nonabelian p-groups, p-groups of maximal class, regular and absolutely regular p-groups, powerful p-groups.


Full text (PDF) (free access)

DOI: 10.3336/gm.41.2.07


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