On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center (Q6612592)

Let \(FG\) be a group algebra of the \(p\)-group \(G\) over a field \(F\) of a characteristic \(p\) and \(H\) is a \(p\)-group . Then the group algebra \(FG\) is called modular. The problem of isomorphism states that isomorphism of modular group algebras \(FG\) and \(FH\) implies isomorphism of the groups \(G\) and \(H\). This problem is solved for arbitrary abelian 2-groups from \textit{W. May} [J. Algebra Appl. 13, No. 4, Article ID 1350125, 14 p. (2014; Zbl 1297.20004)]. This is a fundamental result.\N\NLet \(m(G)\) be the rank of the homocyclic component of G/Z(G) of maximal exponent, i.e., the number of cyclic direct factors of maximal order in this group.\N\NThe authors prove the following results.\N\N``Theorem 1.1. Let \(p\) be an odd prime, \(F\) a field of a characteristic \(p\) and \(G\) a finite p-group of nilpotency class 2 with cyclic center. If \(H\) is a group such that the modular group algebras \(FG\) and \(FH\) are isomorphic, then the groups \(G\) and \(H\) are isomorphic.\N\NTheorem 1.2. Let \(F\) be a field of characteristic 2 and \(G\) a finite 2-group of nilpotency class 2 with cyclic center. Assume moreover that the polynomial \(X^2 +X +1\) is irreducible in the polynomial ring \(F[X]\). If \(H\) is a group such that the modular group algebras \(FG\) and \(FH\) are isomorphic, then the groups \(G\) and \(H\) are isomorphic.\N\NTheorem 1.3. Let \(F\) be a field of characteristic 2 and G a finite 2-group of nilpotency class 2 with cyclic center. If \(FG\) and \(FH\) are isomorphic and \(m(G) \leq 2\), then \(G\) and \(H\) are isomorphic. If \(FG\) and \(FH\) are isomorphic and \(m(G) >2\), then there is exactly one isomorphism type of groups not containing \(G\) to which \(H\) possibly belongs.''