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A note on Friendly and Solitary Groups

Year 2023, Volume: 34 Issue: 34, 197 - 206, 10.07.2023
https://doi.org/10.24330/ieja.1260499

Abstract

In this paper, we extend the notions of friendly and solitary
numbers to group theory and define friendly and solitary groups of
type-1 and type-2. We provide many examples of friendly and
solitary groups and study certain properties of the type-2 friends
of cyclic $p$-groups, where $p$ is a prime number.

References

  • C. W. Anderson, D. Hickerson and M. Greening, Problems and Solutions: Solutions of Advanced Problems: 6020, Friendly integers, Amer. Math. Monthly, 84(1) (1977), 65-66.
  • W. Burnside, On groups of order $p^{\alpha}q^{\beta}$ (Second Paper), Proc. London Math.Soc., 2(2) (1905), 432-437.
  • J. Cemra, 10 Solitary check, Github/CemraJC/Solidarity.
  • W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J.Math., 13 (1963), 775-1029.
  • J. Gallian, Contemporary Abstract Algebra, Chapman and Hall/CRC, 2021.
  • GAP-Groups, Algorithms, and Programming, version 4:12:0, gap-system.org, 2022.
  • K. Guha and S. Ghosh, Measuring abundance with abundancy index, Ball State Undergraduate Math. Exch., 15(1) (2021), 28-39.
  • J. Kirtland, Complementation of Normal Subgroups: in Finite Groups, De Gruyter, Berlin, 2017.
  • T. Leinster, Perfect numbers and groups, arXiv:math/0104012, 2001.
  • T. De Medts and A. Maroti, Perfect numbers and finite groups, Rend. Semin. Mat. Univ. Padova, 129 (2013), 17-33.
  • C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1, Kluwer Academic Publishers, Dordrecht, 2002.
  • G. Mittal and R. K. Sharma, On unit group of finite semisimple group algebrasof non-metabelian groups up to order 72, Math. Bohem., 146(4) (2021), 429-455.
  • P. Pollack and C. Pomerance, Some properties of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B, 3 (2016), 1-26.
  • M. Tarnauceanu, Finite groups determined by an inequality of the orders of their normal subgroups, An. Stiint. Univ. Al. I. Cuza Iai. Mat. (N.S.) 57(2) (2011), 229-238.
Year 2023, Volume: 34 Issue: 34, 197 - 206, 10.07.2023
https://doi.org/10.24330/ieja.1260499

Abstract

References

  • C. W. Anderson, D. Hickerson and M. Greening, Problems and Solutions: Solutions of Advanced Problems: 6020, Friendly integers, Amer. Math. Monthly, 84(1) (1977), 65-66.
  • W. Burnside, On groups of order $p^{\alpha}q^{\beta}$ (Second Paper), Proc. London Math.Soc., 2(2) (1905), 432-437.
  • J. Cemra, 10 Solitary check, Github/CemraJC/Solidarity.
  • W. Feit and J. G. Thompson, Solvability of groups of odd order, Pacific J.Math., 13 (1963), 775-1029.
  • J. Gallian, Contemporary Abstract Algebra, Chapman and Hall/CRC, 2021.
  • GAP-Groups, Algorithms, and Programming, version 4:12:0, gap-system.org, 2022.
  • K. Guha and S. Ghosh, Measuring abundance with abundancy index, Ball State Undergraduate Math. Exch., 15(1) (2021), 28-39.
  • J. Kirtland, Complementation of Normal Subgroups: in Finite Groups, De Gruyter, Berlin, 2017.
  • T. Leinster, Perfect numbers and groups, arXiv:math/0104012, 2001.
  • T. De Medts and A. Maroti, Perfect numbers and finite groups, Rend. Semin. Mat. Univ. Padova, 129 (2013), 17-33.
  • C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, Algebra and Applications, 1, Kluwer Academic Publishers, Dordrecht, 2002.
  • G. Mittal and R. K. Sharma, On unit group of finite semisimple group algebrasof non-metabelian groups up to order 72, Math. Bohem., 146(4) (2021), 429-455.
  • P. Pollack and C. Pomerance, Some properties of Erdos on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B, 3 (2016), 1-26.
  • M. Tarnauceanu, Finite groups determined by an inequality of the orders of their normal subgroups, An. Stiint. Univ. Al. I. Cuza Iai. Mat. (N.S.) 57(2) (2011), 229-238.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Shubham Mıttal This is me

Gaurav Mıttal This is me

R. K. Sharma This is me

Early Pub Date May 11, 2023
Publication Date July 10, 2023
Published in Issue Year 2023 Volume: 34 Issue: 34

Cite

APA Mıttal, S., Mıttal, G., & Sharma, R. K. (2023). A note on Friendly and Solitary Groups. International Electronic Journal of Algebra, 34(34), 197-206. https://doi.org/10.24330/ieja.1260499
AMA Mıttal S, Mıttal G, Sharma RK. A note on Friendly and Solitary Groups. IEJA. July 2023;34(34):197-206. doi:10.24330/ieja.1260499
Chicago Mıttal, Shubham, Gaurav Mıttal, and R. K. Sharma. “A Note on Friendly and Solitary Groups”. International Electronic Journal of Algebra 34, no. 34 (July 2023): 197-206. https://doi.org/10.24330/ieja.1260499.
EndNote Mıttal S, Mıttal G, Sharma RK (July 1, 2023) A note on Friendly and Solitary Groups. International Electronic Journal of Algebra 34 34 197–206.
IEEE S. Mıttal, G. Mıttal, and R. K. Sharma, “A note on Friendly and Solitary Groups”, IEJA, vol. 34, no. 34, pp. 197–206, 2023, doi: 10.24330/ieja.1260499.
ISNAD Mıttal, Shubham et al. “A Note on Friendly and Solitary Groups”. International Electronic Journal of Algebra 34/34 (July 2023), 197-206. https://doi.org/10.24330/ieja.1260499.
JAMA Mıttal S, Mıttal G, Sharma RK. A note on Friendly and Solitary Groups. IEJA. 2023;34:197–206.
MLA Mıttal, Shubham et al. “A Note on Friendly and Solitary Groups”. International Electronic Journal of Algebra, vol. 34, no. 34, 2023, pp. 197-06, doi:10.24330/ieja.1260499.
Vancouver Mıttal S, Mıttal G, Sharma RK. A note on Friendly and Solitary Groups. IEJA. 2023;34(34):197-206.