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October 2021 Classification of finite groups with birdcage-shaped Hasse diagrams
Shigeru TAKAMURA
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Osaka J. Math. 58(4): 885-897 (October 2021).

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 (). 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.

Acknowledgments

We would like to thank Ryota Hirakawa, Takayuki Okuda, and Kenjiro Sasaki for valuable discussions. This work was supported by Grant-in-Aid for Scientific Research(C) Grant Number JP20K03533.

Citation

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Shigeru TAKAMURA. "Classification of finite groups with birdcage-shaped Hasse diagrams." Osaka J. Math. 58 (4) 885 - 897, October 2021.

Information

Received: 13 May 2020; Revised: 15 July 2020; Published: October 2021
First available in Project Euclid: 11 October 2021

Subjects:
Primary: 05C25
Secondary: 20E34

Rights: Copyright © 2021 Osaka University and Osaka City University, Departments of Mathematics

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Vol.58 • No. 4 • October 2021
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