A comparison of group algebras of dihedral and quaternion groups

Leo Creedon; Kieran Hughes; Steve Szabo

doi:10.1007/s00200-020-00485-1

A comparison of group algebras of dihedral and quaternion groups

Leo Creedon
, Kieran Hughes
, Steve Szabo

Eastern Kentucky University

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

Original language	English
Pages (from-to)	245-264
Number of pages	20
Journal	Applicable Algebra in Engineering, Communications and Computing
Volume	32
Issue number	3
DOIs	https://doi.org/10.1007/s00200-020-00485-1
Publication status	Published - Jun 2021

Keywords

Actions of Lie algebras
Dihedral
Group algebra
Group ring
Modular
Quaternion
Reflexive rings
Reversible rings
Semicomutative rings
Symmetric rings

Name of Affiliated ATU Research Unit

MISHE - Mathematical Modelling and Intelligent Systems for Health & Environment

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10.1007/s00200-020-00485-1

Cite this

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title = "A comparison of group algebras of dihedral and quaternion groups",

abstract = "The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.",

keywords = "Actions of Lie algebras, Dihedral, Group algebra, Group ring, Modular, Quaternion, Reflexive rings, Reversible rings, Semicomutative rings, Symmetric rings",

author = "Leo Creedon and Kieran Hughes and Steve Szabo",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.",

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doi = "10.1007/s00200-020-00485-1",

language = "English",

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T1 - A comparison of group algebras of dihedral and quaternion groups

AU - Creedon, Leo

AU - Hughes, Kieran

AU - Szabo, Steve

PY - 2021/6

Y1 - 2021/6

N2 - The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

AB - The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

KW - Actions of Lie algebras

KW - Dihedral

KW - Group algebra

KW - Group ring

KW - Modular

KW - Quaternion

KW - Reflexive rings

KW - Reversible rings

KW - Semicomutative rings

KW - Symmetric rings

UR - https://www.scopus.com/pages/publications/85099577590

U2 - 10.1007/s00200-020-00485-1

DO - 10.1007/s00200-020-00485-1

M3 - Article

AN - SCOPUS:85099577590

SN - 0938-1279

VL - 32

SP - 245

EP - 264

JO - Applicable Algebra in Engineering, Communications and Computing

JF - Applicable Algebra in Engineering, Communications and Computing

IS - 3

ER -

A comparison of group algebras of dihedral and quaternion groups

Abstract

Keywords

Name of Affiliated ATU Research Unit

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