Concerning equally covered groups

Year 2024, Early Access, 1 - 14

T. Foguel A. R. Moghaddamfar J. Schmidt A. Velasquez - Berroteran

https://doi.org/10.24330/ieja.1567377

Abstract

A finite group is equally covered if it has a covering by proper subgroups of equal orders. Among other results, it is shown that finite simple groups have no equal coverings, and for any finite group $G$ the $n^{\text{th}}$ Cartesian power of $G$ has an equal covering for some $n$. Some related topics are also discussed.

Keywords

Equally covered group, Cartesian powers of a group, simple group

References

• R. Atanasov, T. Foguel and A. Penland, Equal quasi-partition of p-groups, Results Math., 64(1-2) (2013), 185-191.
• Y. Berkovich, Minimal nonabelian and maximal subgroups of a finite p-group, Glas. Mat. Ser. III, 43(63)(1) (2008), 97-109.
• Y. Berkovich, Coverings of finite groups by few proper subgroups, Glas. Mat. Ser. III, 45(65)(2) (2010), 415-429.
• M. Bhargava, When is a group the union of proper normal subgroups?, Amer. Math. Monthly, 109(5) (2002), 471-473.
• W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24(3-4) (1997), 235-265.
• M. Bruckheimer, A. C. Bryan and A. Muir, Groups which are the union of three subgroups, Amer. Math. Monthly, 77 (1970), 52-57.
• T. C. Burness and E. Covato, On the prime graph of simple groups, Bull. Aust. Math. Soc., 91(2) (2015), 227-240.
• J. H. E. Cohn, On n-sum groups, Math. Scand., 75(1) (1994), 44-58.
• J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford University Press, 1985.
• T. S. Foguel and M. F. Ragland, Groups with a finite covering by isomorphic abelian subgroups, Computational group theory and the theory of groups, Contemp. Math., 470 (2008), 75-88.
• G. D. Franceschi, Centralizers and conjugacy classes in finite classical groups, Preprint, 2020, https://doi.org/10.48550/arXiv.2008.12651.
• The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.11.1, 2021, https://www.gap-system.org.
• M. A. Grechkoseeva, V. D. Mazurov, W. Shi, A. V. Vasil'ev and N. Yang, Finite groups isospectral to simple groups, Commun. Math. Stat., 11(2) (2023), 169-194.
• S. Haber and A. Rosenfeld, Groups as unions of proper subgroups, Amer. Math. Monthly, 66 (1959), 491-494.
• I. M. Isaacs, Equally partitioned groups, Pacific J. Math., 49 (1973), 109-116.
• I. M. Isaacs, Finite Group Theory, Grad. Stud. Math., 92. American Mathematical Society, Providence, RI, 2008.
• T. Y. Lam, On subgroups of prime index, Amer. Math. Monthly, 111(3) (2004), 256-258.
• M. W. Liebeck, C. E. Praeger and J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. Algebra, 111(2) (1987), 365-383.
• M. W. Liebeck, C. E. Praeger and J. Saxl, Transitive subgroups of primitive permutation groups, J. Algebra, 234(2) (2000), 291-361.
• N. V. Maslova, On the coincidence Grunberg-Kegel graphs of a finite simple group and its proper subgroup, Proc. Steklov Inst. Math., 288 (2015), 129-141.
• A. Pachera, Exponent preserving subgroups of the finite simple groups, Comm. Algebra, 45(6) (2017), 2494-2504.
• D. J. S. Robinson, A Course in the Theory of Groups (Second Edition), Grad. Texts in Math., 80, Springer-Verlag, New York, 1996.
• G. Scorza, I gruppi che possono pensarsi come somme di tre loro sottogruppi, Boll. Unione Mat. Ital., 5(5) (1926), 216-218.
• A. Velasquez-Berroteran, Equal Coverings of Finite Groups, Thesis, https://doi.org/10.48550/arXiv.2206.14843.