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The Modular Isomorphism Problem over all fields

Leo Margolis, Taro Sakurai

TL;DR

The paper analyzes the Modular Isomorphism Problem (MIP) for finite $p$-groups over fields of characteristic $p$, contrasting results over the prime field with those over arbitrary fields. It compiles and leverages $F$-invariants (such as $G/G'$, $Z(G)$, and $D_n(G)/D_{n+1}(G)$) and cohomological data (notably $\dim HH^1(FG)$) to extend several prime-field MIP results to all fields, including finite metacyclic $p$-groups, finite $3$-groups of maximal class, and two-generated class-two groups. The work provides explicit invariant calculations (e.g., type-by-type analyses for maximal-class $3$-groups) and uses transfer lemmas to show that FG $\\cong$ FH implies G $\\cong$ H in most broad classes, identifying a potential exception pattern between certain maximal-class groups ($T_2$ vs $T_3$) when $n$ is even. Overall, it broadens the applicability of MIP techniques across ground fields and clarifies how prime-field proofs can be adapted to more general settings, with concrete invariants guiding the distinctions.

Abstract

The Modular Isomorphism Problem asks, if an isomorphism between modular group algebras of finite $p$-groups over a field $F$ implies an isomorphism of the group bases. We explore the differences of knowledge on the problem when $F$ is either assumed to be a prime field or a general field of characteristic $p$. After revising the literature and explaining reasons for the differences, we generalize some of the positive answers to the problem from the prime field case to the general case.

The Modular Isomorphism Problem over all fields

TL;DR

The paper analyzes the Modular Isomorphism Problem (MIP) for finite -groups over fields of characteristic , contrasting results over the prime field with those over arbitrary fields. It compiles and leverages -invariants (such as , , and ) and cohomological data (notably ) to extend several prime-field MIP results to all fields, including finite metacyclic -groups, finite -groups of maximal class, and two-generated class-two groups. The work provides explicit invariant calculations (e.g., type-by-type analyses for maximal-class -groups) and uses transfer lemmas to show that FG FH implies G H in most broad classes, identifying a potential exception pattern between certain maximal-class groups ( vs ) when is even. Overall, it broadens the applicability of MIP techniques across ground fields and clarifies how prime-field proofs can be adapted to more general settings, with concrete invariants guiding the distinctions.

Abstract

The Modular Isomorphism Problem asks, if an isomorphism between modular group algebras of finite -groups over a field implies an isomorphism of the group bases. We explore the differences of knowledge on the problem when is either assumed to be a prime field or a general field of characteristic . After revising the literature and explaining reasons for the differences, we generalize some of the positive answers to the problem from the prime field case to the general case.
Paper Structure (17 sections, 18 theorems, 32 equations, 1 figure, 4 tables)

This paper contains 17 sections, 18 theorems, 32 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Known results
  3. Classes and invariants
  4. Differences between prime fields and all fields
  5. Difference 1
  6. Difference 2
  7. Difference 3
  8. From prime fields to all fields
  9. Finite metacyclic $p$-groups
  10. Finite $3$-groups of maximal class
  11. Type 1: $C_G(g) = G$
  12. Type 2: $C_G(g) = N$
  13. Type 3: $C_G(g) = \langle g, M \rangle$
  14. Type 4: $C_G(g) = \langle g, Z(G) \rangle$
  15. Two-generated finite $p$-groups of nilpotency class two
  16. ...and 2 more sections

Key Result

Theorem 2.1

Assume $G$ is a finite $p$-group, $F$ a field of characteristic $p$ and $H$ a group such that $FG \cong FH$. If $G$ is among the groups in the following list, then $G \cong H$. Moreover, for $p=2$ in the following cases the statement also holds.

Figures (1)

  • Figure 1: Subgroups in \ref{['prop:TransferLemmaApplications']}. The solid lines depict obtained $F$-invariants.

Theorems & Definitions (29)

  • Theorem 2.1
  • Proposition 2.2: Passman77
  • Theorem 2.3: "Transfer Lemma" GarciaLucas24
  • Proposition 2.4
  • proof
  • Corollary 2.5: GarciaLucas24
  • Proposition 2.6: MargolisSakuraiStanojkovski23
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 19 more