Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 665-670, 2019


On the number of isomorphism classes of derived subgroups

Leyli Jafari Taghvasani, Soran Marzang, Mohammad Zarrin

Received October 7, 2017.   Published online November 29, 2018.

Abstract:  We show that a finite nonabelian characteristically simple group $G$ satisfies $n=|\pi(G)|+2$ if and only if $G\cong A_5$, where $n$ is the number of isomorphism classes of derived subgroups of $G$ and $\pi(G)$ is the set of prime divisors of the group $G$. Also, we give a negative answer to a question raised in M. Zarrin (2014).
Keywords:  derived subgroup; simple group
Classification MSC:  20F24
DOI:  10.21136/CMJ.2018.0464-17


References:
[1] M. J. J. Barry, M. B. Ward: Simple groups contain minimal simple groups. Publ. Mat., Barc. 41 (1997), 411-415. DOI 10.5565/PUBLMAT_41297_07 | MR 1485492 | Zbl 0894.20019
[2] The GAP Group: GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra, Version 4.4. (2005), Available at http://www.gap-system.org. SW 320
[3] F. de Giovanni, D. J. S. Robinson: Groups with finitely many derived subgroups. J. Lond. Math. Soc., II. Ser. 71 (2005), 658-668. DOI 10.1112/S0024610705006484 | MR 2132376 | Zbl 1084.20026
[4] M. Herzog: On finite simple groups of order divisible by three primes only. J. Algebra 10 (1968), 383-388. DOI 10.1016/0021-8693(68)90088-4 | MR 0233881 | Zbl 0167.29101
[5] M. Herzog, P. Longobardi, M. Maj: On the number of commutators in groups. Ischia Group Theory 2004 (Z. Arad et al., eds). Contemporary Mathematics 402, American Mathematical Society, Providence (2006), 181-192. DOI 10.1090/conm/402 | MR 2258662 | Zbl 1122.20017
[6] B. Huppert: Finite Groups I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134, Springer, Berlin (1967). (In German.) DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
[7] M. W. Liebeck, E. A. O'Brien, A. Shalev, P. H. Tiep: The Ore conjecture. J. Eur. Math. Soc. (JEMS) 12 (2010), 939-1008. DOI 10.4171/JEMS/220 | MR 2654085 | Zbl 1205.20011
[8] P. Longobardi, M. Maj, D. J. S. Robinson: Locally finite groups with finitely many isomorphism classes of derived subgroups. J. Algebra 393 (2013), 102-119. DOI 10.1016/j.jalgebra.2013.06.036 | MR 3090061 | Zbl 1294.20049
[9] P. Longobardi, M. Maj, D. J. S. Robinson, H. Smith: On groups with two isomorphism classes of derived subgroups. Glasg. Math. J. 55 (2013), 655-668. DOI 10.1017/S0017089512000821 | MR 3084668 | Zbl 1287.20046
[10] W. Shi: On simple $K_4$-groups. Chin. Sci. Bull. 36 (1991), 1281-1283.
[11] M. Zarrin: On groups with finitely many derived subgroups. J. Algebra Appl. 13 (2014), Article ID 1450045, 5 pages. DOI 10.1142/S0219498814500455 | MR 3200123 | Zbl 1303.20044

Affiliations:   Leyli Jafari Taghvasani, Soran Marzang, Mohammad Zarrin (corresponding author), Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj, Iran, e-mail: L.jafari@sci.uok.ac.ir, S.marzang@sci.uok.ac.ir, M.zarrin@sci.uok.ac.ir


 
PDF available at: