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PROPERTIES OF FINITE GROUPS DETERMINED BY THE PRODUCT OF THEIR ELEMENT ORDERS

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 June 2020

MORTEZA BANIASAD AZAD Open the ORCID record for MORTEZA BANIASAD AZAD [Opens in a new window]  and
BEHROOZ KHOSRAVI Open the ORCID record for BEHROOZ KHOSRAVI [Opens in a new window]
MORTEZA BANIASAD AZAD*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran email baniasad84@gmail.com
BEHROOZ KHOSRAVI
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran email khosravibbb@yahoo.com
*
✉baniasad84@gmail.com
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Abstract

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For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

Keywords

product of element orderssolvable groupssupersolvable groupsnilpotent groupsabelian groupscyclic groups

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D15: Nilpotent groups, $p$-groups 20F16: Solvable groups, supersolvable groups 20F18: Nilpotent groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 88 - 95
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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