Original scientific paper

Short proofs of some basic characterization theorems of finite p-group theory

Yakov Berkovich

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 |A ∩ B| = p, unless p = 2 and G is dihedral.

Keywords

Finite p-groups; metacyclic p-groups; minimal nonabelian p-groups; p-groups of maximal class; regular and absolutely regular p-groups; powerful p-groups

Hrčak ID:

5850

URI

https://hrcak.srce.hr/5850

Publication date:

9.12.2006.

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