On finite groups whose power graph is a cograph
Date
01/02/2022Metadata
Show full item recordAbstract
A P4-free graph is called a cograph. In this paper we partially characterize finite groups whose power graph is a cograph. As we will see, this problem is a generalization of the determination of groups in which every element has prime power order, first raised by Graham Higman in 1957 and fully solved very recently. First we determine all groups G and H for which the power power graph of G times H is a cograph. We show that groups whose power graph is a cograph can be characterised by a condition only involving elements whose orders are prime or the product of two (possibly equal) primes. Some important graph classes are also taken under consideration. For finite simple groups we show that in most of the cases their power graphs are not cographs: the only ones for which the power graphs are cographs are certain groups PSL(2,q) and Sz(q) and the group PSL(3,4). However, a complete determination of these groups involves some hard number-theoretic problems.
Citation
Cameron , P J , Manna , P & Mehatari , R 2022 , ' On finite groups whose power graph is a cograph ' , Journal of Algebra , vol. 591 , pp. 59-74 . https://doi.org/10.1016/j.jalgebra.2021.09.034
Publication
Journal of Algebra
Status
Peer reviewed
ISSN
0021-8693Type
Journal article
Rights
Copyright © 2021 Published by Elsevier. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jalgebra.2021.09.034.
Description
Funding: The author Pallabi Manna is supported by CSIR (Grant No-09/983(0037)/2019-EMR-I). Ranjit Mehatari thanks the SERB, India, for financial support (File Number: CRG/2020/000447) through the Core Research Grant.Collections
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