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On the solvability of a finite group by the sum of subgroup orders

Marius Tarnauceanu
대한수학회 (The Korean Mathematical Society)|2020.11|대한수학회보 (Bulletin of the KMS)|제57권 제6호|pp. 1475-1479|DOI 10.4134/BKMS.b200004

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참고문헌

키워드

solvable groupsSubgroup orders
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초록

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is sol\-va\-ble. This partially solves an open problem posed in \cite{9}.

Abstract

Let $G$ be a finite group and $\sigma_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. Under some restrictions on the number of conjugacy classes of (non-normal) maximal subgroups of $G$, we prove that if $\sigma_1(G)<\frac{117}{20}$, then $G$ is sol\-va\-ble. This partially solves an open problem posed in \cite{9}.

참고문헌 (9)

  1. S. M. J. Amiri, "Characterization of A5 and PSL(2; 7)by sum of element orders", Int. J. Group Theory, 2(2), pp.35~39 (2013)
  2. Morteza Baniasad Azad, "A criterion for solvability of a finite group by the sum of element orders", Journal of Algebra, 516, pp.115~124 (2018), Elsevier BV[DOI]
  3. V A Belonogov, "FINITE GROUPS WITH THREE CLASSES OF MAXIMAL SUBGROUPS", Mathematics of the USSR-Sbornik, 59(1), pp.223~236 (1988), IOP Publishing[DOI]
  4. L. E. Dickson, "Linear groups with an exposition of the Galois field theory, with an intro-duction by W. Magnus" (1958), Dover Publications, Inc
  5. Marcel Herzog, "Two new criteria for solvability of finite groups", Journal of Algebra, 511, pp.215~226 (2018), Elsevier BV[DOI]
  6. T. De Medts, "Finite groups determined by an inequality of the orders of their subgroups", Bull. Belg. Math. Soc. Simon Stevin, 15(4), pp.699~704 (2008)
  7. D. J. S. Robinson, "A Course in the Theory of Groups" (1996), Springer-Verlag
  8. R. Schmidt, "Subgroup Lattices of Groups" (1994), Walter de Gruyter & Co
  9. Marius Tărnăuceanu, "Finite groups determined by an inequality of the orders of their subgroups II", Communications in Algebra, 45(11), pp.4865~4868 (2016), Informa UK Limited[DOI]

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