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Harmonic mean Sylow numbers of nonsolvable groups

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Let G be a finite group. We write hsn(G) for the harmonic mean Sylow number of G. The Fitting subgroup of G is denoted by F(G). It is known that if \(hsn(G)<\frac{45}{7 }\), then G is solvable. In this paper, we extend the study to nonsolvable groups. We prove that if G is a finite nonsolvable group, then the following holds:

  1. (a)

    if \(hsn(G)<\frac{360}{47}\) and \(hsn(G)\ne \frac{480}{71}\), then \( G/F(G)\cong A_{5}\);

  2. (b)

    if \(hsn(G)=\frac{480}{71}\), then \(G/N\cong A_{5}\), where N is the largest normal solvable subgroup of G.

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Correspondence to C. S. Anabanti.

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Communicated by B. Sury.

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Anabanti, C.S., Asboei, A.K. Harmonic mean Sylow numbers of nonsolvable groups. Indian J Pure Appl Math (2024).

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