Covering monolithic groups with proper subgroups

Document Type : Ischia Group Theory 2012


University of Padova


Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is monolithic, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.


Main Subjects

A. Ballester-Bolinches and L. M. Ezquerro (2006). Classes of Finite Groups. Springer, Berlin. A. Abdollahi, F. Ashraf and S. M. Shaker (2007). The symmetric group of degree six can be covered by $13$ and no fewer proper subgroups. Bull. Malays. Math. Sci. Soc. (2). 30 (1), 57-58 A. Abdollahi and S. M. Jafarian Amiri (2008). Minimal coverings of completely reducible groups. Publ. Math. Debrecen. 72 (1-2), 167-172 M. Bhargava (2007). Finiteness criteria for coverings of groups by finitely many subgroups or cosets. Int. Electron. J. Algebra. 2, 83-89 J. H. E. Cohn (1994). On $n$-sum groups. Math. Scand.. 75 (1), 44-58 The GAP Group, GAP -- Groups (2005). Algorithms, and Programming, Version 4.4. (\href{}{}). M. Garonzi (2013). Finite groups that are union of at most $25$ proper subgroups. J. Algebra Appl., 11 pages. 12 (4) M. Garonzi and A. Lucchini (2010). Direct products of groups as unions of proper subgroups. Arch. Math. (Basel). 95 (3), 201-206 M. Garonzi (2013). Covering certain monolithic groups with proper subgroups. Comm. Algebra. 42 (2), 471-491 W. M. Kantor, A. Lubotzky and A. Shalev (2011). Invariable generation and the Chebotarev invariant of a finite group. J. Algebra. 348 (1), 302-314 A. Mar\'oti (2005). Covering the symmetric groups with proper subgroups. J. Combin. Theory Ser. A. 110 (1), 97-111 A. Mar\'oti and M. Garonzi (2011). Covering certain wreath products with proper subgroups. J. Group Theory. 14 (1), 103-125 B. H. Neumann (1954). Groups covered by permutable subsets. J. London Math. Soc.. 29, 236-248 E. Detomi and A. Lucchini (2008). On the structure of primitive $n$-sum groups. Cubo. 10 (3), 195-210 M. J. Tomkinson (1997). Groups as the union of proper subgroups. Math. Scand.. 81 (2), 191-198 M. Garonzi (2012). Coverings of Groups by Subgroups. Ph.D. Thesis, University of Padova, Italy.
Volume 2, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2012
March 2013
Pages 131-144
  • Receive Date: 30 December 2012
  • Revise Date: 08 March 2013
  • Accept Date: 08 March 2013
  • Published Online: 01 March 2013