On a group-theoretical generalization of the Euler’s totient function

Marius Tărnăuceanu ()
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Marius Tărnăuceanu: “Al.I. Cuza” University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 4, 1231-1233

Abstract: Abstract Let G be a finite group and $$\varphi (G)=|\{a\in G \mid o(a)=\exp (G)\}|$$ φ ( G ) = | { a ∈ G ∣ o ( a ) = exp ( G ) } | , where o(a) denotes the order of a in G and $$\exp (G)$$ exp ( G ) denotes the exponent of G. Under a natural hypothesis, in this note we determine the groups G such that $$\forall \, H,K\le G$$ ∀ H , K ≤ G , $$H\subseteq K$$ H ⊆ K implies $$\varphi (H)\mid \varphi (K)$$ φ ( H ) ∣ φ ( K ) . This partially answers Problem 5.4 in [6].

Keywords: Euler’s totient function; Finite group; Order of an element; Exponent of a group; Nilpotent group; Schmidt group; Primary 20D60; 11A25; Secondary 20D99; 11A99 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00429-8

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