On the modular isomorphism problem for 2-generated groups with cyclic derived subgroup

We continue the analysis of the modular isomorphism problem for 2-generated p-groups with cyclic derived subgroup, p > 2, started in [D. Garcia-Lucas, A del Rio and M. Stanojkowski, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebra Represent. Theory 26 (2022) 2683-2707, doi:10.1007/s10468-022-10182-x]. We show that if G belongs to this class of groups, then the isomorphism type of the quotients G/(G ')(p3) and G/gamma(3)(G)(p) are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification [O. Broche, D. Garcia-Lucas and A. del Rio, A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order, Int. J. Algebra Comput. 33(4) (2023) 641-686]. We also show that for groups in this class of order at most p(11), the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order p(12) whose group algebras over the field with p elements cannot be distinguished with the techniques available to us.