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GROUPS WITH A GIVEN NUMBER OF NONPOWER SUBGROUPS

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  10 January 2022

C. S. ANABANTI Open the ORCID record for C. S. ANABANTI [Opens in a new window]  and
S. B. HART Open the ORCID record for S. B. HART [Opens in a new window]
C. S. ANABANTI*
Affiliation:
Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria 0002, South Africa
S. B. HART
Affiliation:
Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet St, London WC1E 7HX, UK e-mail: s.hart@bbk.ac.uk
*
e-mail: chimere.anabanti@up.ac.za
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Abstract

No group has exactly one or two nonpower subgroups. We classify groups containing exactly three nonpower subgroups and show that there is a unique finite group with exactly four nonpower subgroups. Finally, we show that given any integer k greater than $4$, there are infinitely many groups with exactly k nonpower subgroups.

Keywords

counting subgroupsnonpower subgroupsfinite groups

MSC classification

Primary: 20D25: Special subgroups (Frattini, Fitting, etc.)
Secondary: 20D60: Arithmetic and combinatorial problems 20E34: General structure theorems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 315 - 319
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Anabanti, C. S., Aroh, A. B., Hart, S. B. and Oodo, A. R., ‘A question of Zhou, Shi and Duan on nonpower subgroups of finite groups’, Quaest. Math. (2021); doi:10.2989/16073606.2021.1924891.Google Scholar
Berkovich, Y., Groups of Prime Power Order, Volume 1, De Gruyter Expositions in Mathematics, 46 (De Gruyter, Berlin, 2008).Google Scholar
Zhou, W., Shi, W. and Duan, Z., ‘A new criterion for finite noncyclic groups’, Comm. Algebra 34 (2006), 4453–4457.CrossRefGoogle Scholar