[Paper Review] On the isomorphism problem for unit groups of modular group algebras

Alexander Konovalov, A. G. Krivokhata|arXiv (Cornell University)|Jul 12, 2006

Finite Group Theory Research3 references3 citations

TL;DR

This paper confirms the Unit Group Isomorphism Problem (UMIP) for all 2-groups of order up to 32 by computationally verifying that their normalized unit groups in modular group algebras over the field of two elements are pairwise non-isomorphic. Using GAP and the LAGUNA package, the authors compute invariants such as the order of the Frattini subgroup, number of involutions, automorphism group structure, and center exponent to distinguish unit groups, ultimately proving that each 2-group of order ≤32 is uniquely determined by its normalized unit group.

ABSTRACT

Using the computational algebra system GAP (http://www.gap-system.org) and the GAP package LAGUNA (http://www.cs.st-andrews.ac.uk/~alexk/laguna.htm), we checked that all 2-groups of order not greater than 32 are determined by normalized unit groups of their modular group algebras over the field of two elements.

Motivation & Objective

• To investigate whether finite 2-groups of order up to 32 are uniquely determined by the normalized unit group of their modular group algebra over 𝔽₂.
• To test the strength of the Unit Group Isomorphism Problem (UMIP) as a stronger variant of the Modular Isomorphism Problem (MIP).
• To apply computational algebra techniques to distinguish non-isomorphic normalized unit groups of modular group algebras.
• To provide a computational verification of UMIP for 2-groups of order 16 and 32, extending known results for abelian and maximal class 2-groups.

Proposed method

• Employed the computational algebra system GAP with the LAGUNA package to compute power-commutator presentations of normalized unit groups V(KG) for modular group algebras KG over 𝔽₂.
• Used 'cheap' invariants such as the order of the Frattini subgroup and center of V(KG) to initially partition groups into families.
• Calculated the number of involutions in V(KG) as a primary distinguishing invariant for non-isomorphic unit groups.
• Applied the AutPGrp package to compute automorphism group structures, using orders and minimal number of generators to resolve remaining ambiguous pairs.
• For groups of order 32, used coset enumeration due to GAP limitations, processing elements in cosets of proper subgroups to manage memory and computation time.
• Validated results by comparing with theoretical results on 2-groups of maximal class and by cross-checking with known invariants from prior literature.

Experimental results

Research questions

• RQ1 Are all 2-groups of order ≤32 uniquely determined by the isomorphism type of their normalized unit group V(KG) in the modular group algebra over 𝔽₂?
• RQ2 Can computational invariants such as the number of involutions and automorphism group structure effectively distinguish non-isomorphic normalized unit groups of modular group algebras?
• RQ3 Does the Unit Group Isomorphism Problem (UMIP) hold for all 2-groups of order 16 and 32, given that it is known to hold for abelian and maximal class 2-groups?
• RQ4 Can the computational approach using GAP and LAGUNA be systematically applied to verify UMIP for larger classes of p-groups?
• RQ5 What invariants are most effective in resolving the most difficult pairs of non-isomorphic 2-groups with isomorphic normalized unit groups?

Key findings

• All 2-groups of order 16 were confirmed to be determined by their normalized unit groups V(KG), with the order of the Frattini subgroup and number of involutions in V(KG) sufficient to distinguish all non-isomorphic pairs.
• For 2-groups of order 32, the authors identified 12 families using invariants like center and Frattini subgroup orders, and resolved all 41 non-abelian, non-maximal class groups using higher-order invariants.
• The number of involutions in V(KG) successfully distinguished 10 of the 12 families, with 28 groups resolved solely by this invariant.
• For the most challenging pair (G₁₃ and G₁₄), the minimal number of generators of Aut(V(KG)) was used—15 for G₁₃ and 16 for G₁₄—to establish non-isomorphism.
• The automorphism group orders of V(KG) provided decisive distinctions in several cases, such as families 1 and 2, where |Aut(V(KG))| was 2¹⁰² and 2¹⁰¹ for groups 43 and 44.
• The computation for one unit group of order 32 required approximately 24 hours on a high-performance cluster, highlighting the computational intensity of the method.