# Article

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Keywords:
non-perfect group; \$G\$-conjugacy class; \$n\$-decomposable group
Summary:
Let \$G\$ be a finite group. A normal subgroup \$N\$ of \$G\$ is a union of several \$G\$-conjugacy classes, and it is called \$n\$-decomposable in \$G\$ if it is a union of \$n\$ distinct \$G\$-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5.
References:
[1] Ashrafi, A. R.: On decomposability of finite groups. J. Korean Math. Soc. 41 (2004), 479-487. DOI 10.4134/JKMS.2004.41.3.479 | MR 2050157 | Zbl 1058.20026
[2] Ashrafi, A. R., Sahraei, H.: On finite groups whose every normal subgroup is a union of the same number of conjugacy classes. Vietnam J. Math. 30 (2002), 289-294. MR 1933567 | Zbl 1018.20026
[3] Ashrafi, A. R., Venkataraman, G.: On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes. Proc. Indian Acad. Sci., Math. Sci. 114 (2004), 217-224. DOI 10.1007/BF02830000 | MR 2083462 | Zbl 1070.20027
[4] Gorenstein, D.: Finite Groups. Chelsea Publishing Company, New York (1980). MR 0569209 | Zbl 0463.20012
[5] Guo, X., Chen, R.: On finite \$X\$-decomposable groups for \$X=\{1, 2, 3, 4\}\$. Bull. Iranian Math. Soc. 40 (2014), 1243-1262. MR 3273835 | Zbl 06572891
[6] Guo, X. Y., Li, J., Shum, K. P.: On finite \$X\$-decomposable groups for \$X=\{1, 2, 4\}\$. Sib. Math. J. 53 (2012), 444-449 translation from\global\questionmarktrue Sib. Mat. Zh. 53 558-565 2012. DOI 10.1134/S0037446612020255 | MR 2978574 | Zbl 1257.20031
[7] Isaacs, I. M.: Character Theory of Finite Groups. Dover Publications, New York (1994). MR 1280461 | Zbl 0849.20004
[8] Riese, U., Shahabi, M. A.: Subgroups which are the union of four conjugacy classes. Commun. Algebra 29 (2001), 695-701. DOI 10.1081/AGB-100001534 | MR 1841992 | Zbl 0990.20020
[9] Rose, H. E.: A Course on Finite Groups. Universitext, Springer, London (2009). DOI 10.1007/978-1-84882-889-6 | MR 2583713 | Zbl 1200.20001
[10] Shi, W. J.: A class of special minimal normal subgroups. J. Southwest Teachers College 9 (1984), 9-13 Chinese.
[11] Wang, J.: A special class of normal subgroups. J. Chengdu Univ. Sci. Technol. 1987 (1987), 115-119 Chinese. English summary. MR 1028900 | Zbl 0671.20022

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