Skip to main content Accessibility help
×
Home
Hostname: page-component-5d6d958fb5-ls6xp Total loading time: 0.177 Render date: 2022-11-26T17:39:52.885Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

SOME CRITERIA FOR SOLVABILITY AND NILPOTENCY OF FINITE GROUPS BY STRONGLY MONOLITHIC CHARACTERS

Published online by Cambridge University Press:  22 September 2022

SULTAN BOZKURT GÜNGÖR
Affiliation:
Department of Mathematics, Faculty of Science, Istanbul University, 34134Istanbul, Turkey e-mail: sultan__bozkurt@hotmail.com
TEMHA ERKOÇ*
Affiliation:
Department of Mathematics, Faculty of Science, Istanbul University, 34134Istanbul, Turkey

Abstract

Gagola and Lewis [‘A character theoretic condition characterizing nilpotent groups’, Comm. Algebra27 (1999), 1053–1056] proved that a finite group G is nilpotent if and only if $\chi (1)^{2}$ divides $\lvert G:\textrm {ker}\,\chi \rvert $ for every irreducible character $\chi $ of G. The theorem was later generalised by using monolithic characters. We generalise the theorem further considering only strongly monolithic characters. We also give some criteria for solvability and nilpotency of finite groups by their strongly monolithic characters.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The work of the authors was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), project number 119F295.

References

Berkovich, Y. G., ‘On Isaacs’ three character degrees theorem’, Proc. Amer. Math. Soc. 125(3) (1997), 669677.CrossRefGoogle Scholar
Berkovich, Y. G. and Zhmud, E. M., Characters of Finite Groups. Part II, Translations of Mathematical Monographs, 181 (American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
Erkoç, T., Güngör, S. B. and Özkan, J. M., ‘Strongly monolithic characters of finite groups’, J. Algebra Appl., to appear. Published online (28 May 2022).CrossRefGoogle Scholar
Gagola, S. M. and Lewis, M. L., ‘A character theoretic condition characterizing nilpotent groups’, Comm. Algebra 27 (1999), 10531056.CrossRefGoogle Scholar
Isaacs, I. M., Character Theory of Finite Groups (Academic Press, New York, 1976).Google Scholar
Isaacs, I. M., ‘Large orbits in actions of nilpotent groups’, Proc. Amer. Math. Soc. 127(1) (1999), 4550.CrossRefGoogle Scholar
Isaacs, I. M., Algebra: A Graduate Course, Graduate Studies in Mathematics, 100 (American Mathematical Society, Providence, RI, 2009).CrossRefGoogle Scholar
Isaacs, I. M. and Knutson, G., ‘Irreducible character degrees and normal subgroups’, J. Algebra 199(1) (1998), 302326.CrossRefGoogle Scholar
Isaacs, I. M. and Passman, D. S., ‘A characterization of groups in terms of the degrees of their characters II’, Pacific J. Math. 24(3) (1968), 467510.CrossRefGoogle Scholar
Lu, J., Qin, X. and Liu, X., ‘Generalizing a theorem of Gagola and Lewis characterizing nilpotent groups’, Arch. Math. 108 (2017), 337339.CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

SOME CRITERIA FOR SOLVABILITY AND NILPOTENCY OF FINITE GROUPS BY STRONGLY MONOLITHIC CHARACTERS
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

SOME CRITERIA FOR SOLVABILITY AND NILPOTENCY OF FINITE GROUPS BY STRONGLY MONOLITHIC CHARACTERS
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

SOME CRITERIA FOR SOLVABILITY AND NILPOTENCY OF FINITE GROUPS BY STRONGLY MONOLITHIC CHARACTERS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *