The modular isomorphism problem for finite p-groups with a cyclic subgroup of index p²

Summary

Abstract

Let p be a prime number, G be a finite p-group and K be a field of characteristic p. The Modular Isomorphism Problem (MIP) asks whether the group algebra KG determines the group G. Dealing with MIP, we investigated a question whether the nilpotency class of a finite p-group is determined by its modular group algebra over the field of p elements. We give a positive answer to this question provided one of the following conditions holds: (i) exp G = p; (ii) c1(G) = 2; (iii) G' is cyclic; (iv) G is a group of maximal class and contains an abelian subgroup of index p.

As a consequence, the positive solution of MIP for all p-groups containing a cyclic subgroup of index p² was obtained.

Introduction

Though the Modular Isomorphism Problem is known for more than 50 years, up to now it remains open. It was solved only for some classes of p-groups, in particular:

• abelian p-groups (Deskins [10]; alternate proof by Coleman [9]);

• p-groups of class 2 with elementary abelian commutator subgroup (Sandling, theorem 6.25 in [22]);

• metacyclic p-groups (for p > 3 by Bagiński [1]; completed by Sandling [24])

• 2-groups of maximal class (Carlson [8]; alternate proof by Bagiński [2]);

• p-groups of maximal class, p≠2, when |G| ≤ p^p+1 and G contains an abelian maximal subgroup (Caranti and Bagiński [4]);

• elementary abelian-by-cyclic groups (Bagiński [3]);

• p-groups with the center of index p² (Drensky [11]),