Influence of complemented subgroups on the structure of finite groups

Document Type : Research Paper


Faculty of Mathematics, University of Bialystok, 15-245 Bialystok, Ciolkowskiego 1M, Poland


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.


[1] Z. Arad and E. Fisman, On finite factorizable groups, J. Algebra, 86 (1984) 522–548.
[2] A. Ballester-Bolinches, R. Esteban-Romero and D. J. S. Robinson, On finite minimal non-nilpotent groups, Proc.
Amer. Math. Soc., 133 (2005) 3455–3462.
[3] A. Ballester-Bolinches, R. Esteban-Romero and Lu Jiakuan, On finite groups with many supersoluble subgroups,
Arch. Math. (Basel), 109 (2017) 3–8.
[4] A. Ballester-Bolinches and X. Guo, On complemented subgroups of finite groups, Arch. Math. (Basel), 72 (1999)
[5] T. Connor and D. Leemans, An atlas of subgroup lattices of finite almost simple groups, arXiv:1306.4820., (2013)
pp. 6.
[6] L. E. Dickson, Linear groups. With an exposition of the Galois field theory, Dover Publications Inc, New York, 1958.
[7] X. Guo, K. P. Shum and A. Ballester-Bolinches, On complemented minimal subgroups in finite groups, J. Group
Theory, 6 (2003) 159–167.
[8] P. Hall, Complemented groups, J. London Math. Soc., 12 (1937) 201–204.
[9] B. Huppert, Endliche Gruppen I., Springer, Berlin, 1967.
[10] N. Ito, On the factorizations of the linear fractional group LF(2, pn), Acta Sci. Math. (Szeged), 15 (1953) 79–84.
[11] Y. Li, N. Su and Y. Wang, Complemented subgroups and the structure of finite groups, Monatsh. Math., 173 (2014)
[12] S. Li and Y. Zhao, Some finite nonsolvable groups characterized by their solvable subgroups, Acta Math. Sinica
(N.S.), 4 (1988) 5–13.
[13] V. S. Monakhov and V. N. Kniahina, On the Derived Length of a Finite Group with Complemented Subgroups of
Order p2, Ukrainian Math. J., 67 (2015) 989–997.
[14] J. S. Rose, A course on group theory., Cambridge University Press, Cambridge, (1978).
[15] M. Suzuki, A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U. S. A., 46 (1960) 868–870.
[16] M. Suzuki, Group theory I, Springer, Berlin, 1982.
[17] J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 74
(1968) 383–437.
[18] R. A. Wilson, The finite simple groups, Springer–Verlag London Ltd., London, 2009.
  • Receive Date: 10 September 2019
  • Revise Date: 13 December 2019
  • Accept Date: 16 December 2019
  • Published Online: 01 June 2021