فهرست مطالب
International Journal of Group Theory
Volume:10 Issue: 2, Jun 2021
- تاریخ انتشار: 1399/10/25
- تعداد عناوین: 5
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Pages 55-60
Let $ G $ be a finite group and ( psi(G)=sum_{gin G}o(g) ), where $ o(g) $ denotes the order of $gin G$. We show that the Conjecture 4.6.5 posed in [Group Theory and Computation, (2018) 59-90], is incorrect. In fact, we find a pair of finite groups $G$ and $S$ of the same order such that $ psi(G)<psi(S)$, with $G$ solvable and $S$ simple.
* The formulas are not displayed correctly.Keywords: Finite group, element orders sum, solvable group, simple group -
Pages 61-64In this note we prove an analog of Hilbert's theorem 90 for finite nilpotent groups. Our version of Hilbert's theorem 90 was inspired by the Boston--Bush--Hajir (BBH) heuristics in number theory and will be useful in extending the BBH heuristics beyond quadratic field extensions.Keywords: Nilpotent groups, BBH Heuristics, Hilbert's Theorem 90
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Pages 65-74
P. Hall proved that a finite group $G$ is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of $G$ is complemented in $G$. He called such groups complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-complemented groups. We give the classification of finite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every finite group with less than $21$ non-complemented non-minimal ${2,3,5}$-subgroups is soluble.
* The formulas are not displayed correctly.Keywords: Complemented subgroups, Complemented groups, Minimal non-complemented groups, Finite groups -
Pages 75-88
The effect of restricting the set of primes dividing the orders of the finite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally finite groups. The connection with the extraction of roots is explored.
* The formulas are not displayed correctly.Keywords: Finite quotients, Prime divisors -
Pages 89-95
For a given odd prime $p$, we investigate the power graphs of three classes of finite groups: the elementary abelian groups of exponent $p$, and the extra special groups of exponents $p$ or $p^2$. We show that these power graphs are Eulerian for every $p$. As a corollary, we describe two classes of non-isomorphic groups with isomorphic power graphs. In addition, we prove that the clique graphs of the power graphs of two considered classes are complete.
* The formulas are not displayed correctly.Keywords: Power graphs, non-group semigroups, semidirect product
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