Research Article
Year 2023,
Volume: 34 Issue: 34, 197 - 206, 10.07.2023
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.
Keywords
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
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 | |
| Early Pub Date | May 11, 2023 |
| Publication Date | July 10, 2023 |
| Published in Issue | Year 2023 Volume: 34 Issue: 34 |