Existence problems in Hopf-Galois structures and skew braces

2022 Fiscal Year Final Research Report

• 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 (TSANGCINDY) 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
• 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

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.

Free Research Field

群論

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．