CLASSIFICATION OF FINITE GROUPS WITH BIRDCAGE-SHAPED HASSE DIAGRAMS

• Takamura Shigeru

Abstract

The subgroup posets of finite groups are illustrated by graphs (Hasse diagrams). The properties of these graphs have been studied by many researchers — initiated by K. Brown and D. Quillen for p-subgroups. We consider the opposite direction, that is, a realization problem: Given a graph, when does a finite group with its Hasse diagram being the graph exist?, and if any, classify all such finite groups. The Hasse diagram of a group is not arbitrary — it has the top and the bottom vertices, and they are connected by paths of edges. We divide such graphs into two types “branched” and “unbranched”, where unbranched graphs are birdcage-shaped, and finite groups with their Hasse diagrams being such graphs are called birdcage groups. We completely classify the unbranched case: A birdcage group is either a cyclic group of prime power order or a semidirect product of two cyclic groups of prime orders (the orders are possibly equal). In the former, the Hasse diagram is a straight line (a birdcage with a single bar) and in the latter, a birdcage with all bars being of length 2.

Journal

• Osaka Journal of Mathematics

  Osaka Journal of Mathematics 58 (4), 885-897, 2021-10

  Osaka University and Osaka City University, Departments of Mathematics

Keywords

• 05C25
• 20E34

Details

• CRID: 1390853651111161472
• NII Article ID: 120007163111
• NII Book ID: AA00765910
• DOI: 10.18910/84955
• HANDLE: 11094/84955
• ISSN: 00306126
• Web Site: https://dlisv03.media.osaka-cu.ac.jp/il/meta_pub/G0000438repository_00306126-58-4-885
• Text Lang: en
• Data Source:
  • JaLC
  • IRDB
  • CiNii Articles
  • KAKEN