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.
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 ; alternate proof by Coleman );
p-groups of class 2 with elementary abelian commutator subgroup (Sandling, theorem 6.25 in );
metacyclic p-groups (for p > 3 by Bagiński ; completed by Sandling )
2-groups of maximal class (Carlson ; alternate proof by Bagiński );
p-groups of maximal class, p≠2, when |G| ≤ pp+1 and G contains an abelian maximal subgroup (Caranti and Bagiński );
elementary abelian-by-cyclic groups (Bagiński );
p-groups with the center of index p2 (Drensky ),
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