Existence problems in Hopf-Galois structures and skew braces

• Project/Area Number: 21K20319
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Ochanomizu University
• Principal Investigator: TSANG SINYI お茶の水女子大学, 基幹研究院, 助教 (10908271)
• Project Period (FY): 2021-08-30 – 2023-03-31
• Project Status: Completed (Fiscal Year 2022)
• Budget Amount: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: holomorph / 正則部分群 / ホップ・ガロア構造 / skew brace / 有限単純群 / p-groups of class two / multiple holomorph / ホップ・ガロワ構造 / $p$-groups of class two / 巡回拡大 / 巡回群

Outline of Research at the Start

有限群Nのholomorphに含まれる正則部分群Gは，ホップ・ガロワ構造およびskew braceという代数的構造と対応していることが知られている．本研究では，GとNの同型類がどのように関係しているかを調べる．特に，Gが巡回群または概単純群である場合に着目し，Nの同型類を全て特定しようと試みる．さらに，Gが可解群でありNが非可解群となる例が限られているのではないかと思われ，この現象についても深く掘り下げたいと考えている．

Outline of Final Research Achievements

For any finite groups G and N of the same order, let us say that (G,N) is realizable when the holomorph of N contains a regular subgroup isomorphic to G. In this research, for any cyclic group G, we were able to determine the groups N for which (G,N) is realizable. Moreover, we were able to determine the simple groups N for which there exists a solvable group G such that (G,N) is realizable. As a related problem, we also studied the so-called multiple holomorph of some p-groups of class two.

Academic Significance and Societal Importance of the Research Achievements

同位数をもつ有限群GとNに対して，(G,N)がrealizableであることは，ガロア群Gをもつ拡大にタイプNのホップ・ガロア構造が存在すること，及び加法群がNで乗法群がGとなるようなskew braceが存在することと同値である．前者は整数環のガロア加群構造の研究に応用があり，後者はYang-Baxter方程式の集合理論的解と関連していることが知られている．よって，(G,N)がrealizableか否かは重要な問題であり，本研究の成果はこのrealizabilityに関する研究を進展させた．

Research Products

• [Int'l Joint Research] University of Trento(イタリア)
• [Journal Article] Finite $p$-groups of class two with a large multiple holomorph (2023)
  • Author(s): Caranti A.、Tsang Cindy (Sin Yi)
  • Journal Title: Journal of Algebra Volume: 617 Pages: 476-499
  • DOI: 10.1016/j.jalgebra.2022.11.013
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group (2022)
  • Author(s): Tsang Cindy (Sin Yi)
  • Journal Title: Proceedings of the American Mathematical Society, Series B Volume: 9 Issue: 36 Pages: 377-392
  • DOI: 10.1090/bproc/138
  • Peer Reviewed / Open Access
• [Presentation] Regular subgroups in the holomorph, fixed point free pairs of homomorphisms, and factorizations of groups (2023)
  • Author(s): Tsang Cindy (Sin Yi)
  • Organizer: Hopf Algebras & Galois Module Theory
  • Int'l Joint Research
• [Presentation] Non-abelian simple groups which can occur as the additive group of a skew brace with solvable multiplicative group (2022)
  • Author(s): Tsang Cindy (Sin Yi)
  • Organizer: Oberwolfach Mini-Workshop: Skew Braces and the Yang-Baxter Equation
  • Int'l Joint Research
• [Presentation] Finite $p$-groups of class two with a very large multiple holomorph (2022)
  • Author(s): Tsang Cindy (Sin Yi)
  • Organizer: Hopf Algebras & Galois Module Theory
  • Int'l Joint Research
• [Presentation] Characterization of the type of Hopf-Galois structures on cyclic extensions (2022)
  • Author(s): Tsang Cindy (Sin Yi)
  • Organizer: Hopf Algebras & Galois Module Theory
  • Int'l Joint Research
• [Presentation] Characterization of the type of Hopf-Galois structures on cyclic extensions (2022)
  • Author(s): TSANG SIN YI
  • Organizer: Hopf Algebras and Galois Module Theory 2022
  • Int'l Joint Research