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ON THE CHARACTERISATION OF ALTERNATING GROUPS BY CODEGREES

Part of: Representation theory of groups

Published online by Cambridge University Press:  26 January 2024

MALLORY DOLORFINO Open the ORCID record for MALLORY DOLORFINO [Opens in a new window] ,
LUKE MARTIN Open the ORCID record for LUKE MARTIN [Opens in a new window] ,
ZACHARY SLONIM Open the ORCID record for ZACHARY SLONIM [Opens in a new window] ,
YUXUAN SUN Open the ORCID record for YUXUAN SUN [Opens in a new window]  and
YONG YANG Open the ORCID record for YONG YANG [Opens in a new window]
MALLORY DOLORFINO
Affiliation:
Department of Mathematics, Kalamazoo College, Kalamazoo, Michigan, USA e-mail: mallory.dolorfino19@kzoo.edu
LUKE MARTIN
Affiliation:
Department of Mathematics, Gonzaga University, Spokane, Washington, USA e-mail: lwmartin2019@gmail.com
ZACHARY SLONIM
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California, USA e-mail: zachslonim@berkeley.edu
YUXUAN SUN
Affiliation:
Department of Mathematics and Statistics, Haverford College, Haverford, Pennsylvania, USA e-mail: ysun1@haverford.edu
YONG YANG*
Affiliation:
Department of Mathematics, Texas State University, San Marcos, Texas, USA
*
e-mail: yang@txstate.edu
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Abstract

Let G be a finite group and $\mathrm {Irr}(G)$ the set of all irreducible complex characters of G. Define the codegree of $\chi \in \mathrm {Irr}(G)$ as $\mathrm {cod}(\chi ):={|G:\mathrm {ker}(\chi ) |}/{\chi (1)}$ and let $\mathrm {cod}(G):=\{\mathrm {cod}(\chi ) \mid \chi \in \mathrm {Irr}(G)\}$ be the codegree set of G. Let $\mathrm {A}_n$ be an alternating group of degree $n \ge 5$. We show that $\mathrm {A}_n$ is determined up to isomorphism by $\operatorname {cod}(\mathrm {A}_n)$.

Keywords

codegreealternating group

MSC classification

Primary: 20D06: Simple groups
Secondary: 20C15: Ordinary representations and characters
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 6
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230-21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim and Sun during the Summer of 2022 under the supervision of Yang. Yang was also partially supported by a grant from the Simons Foundation (#918096).

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