Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A DENSITY THEOREM RELATED TO DIHEDRAL GROUPS

  • Arya Chandran (Department of Mathematics Institute of Science and Technology Chinmaya Vishwa Vidyapeeth) ;
  • Kesavan Vishnu Namboothiri (Department of Mathematics Baby John Memorial Government College and Department of Collegiate Education Government of Kerala) ;
  • Vinod Sivadasan (Department of Mathematics College of Engineering Trivandrum)
  • Received : 2023.05.04
  • Accepted : 2023.12.05
  • Published : 2024.05.31
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Abstract

For a finite group G, let 𝜓(G) denote the sum of element orders of G. If ${\psi}^{{\prime}{\prime}}(G)\,=\,{\frac{\psi(G)}{{\mid}G{\mid}^2}}$, we show here that the image of 𝜓'' on the class of all Dihedral groups whose order is twice a composite number greater than 4 is dense in $[0,\,{\frac{1}{4}}]$. We also derive some properties of 𝜓'' on the class of all dihedral groups whose order is twice a prime number.

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Acknowledgement

The authors thank the reviewer for offering valuable comments which made this paper more accurate than what it was before.

References

  1. H. Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187-190. https://doi.org/10.1142/S0219498811004057 
  2. H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups, Comm. Algebra 37 (2009), no. 9, 2978-2980. https://doi.org/10.1080/00927870802502530 
  3. T. M. Apostol, Calculus. Vol. I, second edition, Blaisdell Publishing Co., Waltham, MA, 1967. 
  4. M. Herzog, P. Longobardi, and M. Maj, An exact upper bound for sums of element orders in non-cyclic finite groups, J. Pure Appl. Algebra 222 (2018), no. 7, 1628-1642. https://doi.org/10.1016/j.jpaa.2017.07.015 
  5. M. Herzog, P. Longobardi, and M. Maj, Sums of element orders in groups of order 2m with m odd, Comm. Algebra 47 (2019), no. 5, 2035-2048. https://doi.org/10.1080/00927872.2018.1527924 
  6. S. M. Jafarian Amiri and M. Amiri, Second maximum sum of element orders on finite groups, J. Pure Appl. Algebra 218 (2014), no. 3, 531-539. https://doi.org/10.1016/j.jpaa.2013.07.003 
  7. S. M. Jafarian Amiri and M. Amiri, Characterization of p-groups by sum of the element orders, Publ. Math. Debrecen 86 (2015), no. 1-2, 31-37. https://doi.org/10.5486/PMD.2015.5961 
  8. M. Jahani, Y. Marefat, H. Refaghat, and B. V. Fasaghandisi, The minimum sum of element orders of finite groups, Int. J. Group Theory 10 (2021), no. 2, 55-60. 
  9. M.-S. Lazorec and M. Tarnauceanu, A density result on the sum of element orders of a finite group, Arch. Math. (Basel) 114 (2020), no. 6, 601-607. https://doi.org/10.1007/s00013-020-01437-4 
  10. Z. Nitecki, Cantorvals and subsum sets of null sequences, Amer. Math. Monthly 122 (2015), no. 9, 862-870. https://doi.org/10.4169/amer.math.monthly.122.9.862 
  11. R. Shen, G. Chen, and C. Wu, On groups with the second largest value of the sum of element orders, Comm. Algebra 43 (2015), no. 6, 2618-2631. https://doi.org/10.1080/00927872.2014.900686 
  12. W. A. Sutherland, Introduction to Metric and Topological Spaces, Oxford Univ. Press, Oxford, 2009. 
  13. M. Tarnauceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. 238 (2020), no. 2, 629-637. https://doi.org/10.1007/s11856-020-2033-9 
  14. M. Tarnauceanu and D. G. Fodor, On the sum of element orders of finite abelian groups, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 60 (2014), no. 1, 1-7. https://doi.org/10.2478/aicu-2013-0013