Finite groups with many cyclic subgroups

Xiaofang Gao and Rulin Shen ()
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Xiaofang Gao: Hubei Minzu University
Rulin Shen: Hubei Minzu University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 2, 485-498

Abstract: Abstract Let G be a finite group, c(G) the number of its cyclic subgroups, and $$\alpha (G)=c(G)/|G|$$ α ( G ) = c ( G ) / | G | . Set $$I(G)=|\{g\in G|g^2=1\}|$$ I ( G ) = | { g ∈ G | g 2 = 1 } | . In this paper we prove if $$\alpha (G)=3/4$$ α ( G ) = 3 / 4 , then G is isomorphic to a direct product of an elementary abelian 2-group and a dihedral group $$D_{16}, D_{24}$$ D 16 , D 24 , or a group satisfying $$I({G})=\frac{1}{2}|{G}|$$ I ( G ) = 1 2 | G | and $$\exp ({G})=4$$ exp ( G ) = 4 .

Keywords: Number of cyclic subgroups; 2-Groups; Involutions; 20D60; 20D06 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00270-5

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