Kaheni, A., Johari, F. (2019). Characterizing some groups with nilpotent derived subgroup. Algebraic Structures and Their Applications, 6(1), 55-63. doi: 10.29252/as.2019.1353

Azam Kaheni; Farangis Johari. "Characterizing some groups with nilpotent derived subgroup". Algebraic Structures and Their Applications, 6, 1, 2019, 55-63. doi: 10.29252/as.2019.1353

Kaheni, A., Johari, F. (2019). 'Characterizing some groups with nilpotent derived subgroup', Algebraic Structures and Their Applications, 6(1), pp. 55-63. doi: 10.29252/as.2019.1353

Kaheni, A., Johari, F. Characterizing some groups with nilpotent derived subgroup. Algebraic Structures and Their Applications, 2019; 6(1): 55-63. doi: 10.29252/as.2019.1353

Characterizing some groups with nilpotent derived subgroup

^{1}Department of Mathematics, University of Birjand, Birjand, Iran.

^{2}Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.

Abstract

In this paper, groups with trivial intersection between Frattini and derived subgroups are considered. First, some structural properties of these groups are given in an important special case. Then, some family invariants of each $n$-isoclinism family of such groups are stated. In particular, an explicit bound for the order of each center factor group in terms of the order of its derived subgroup is also provided.

[1] X. Guo and L. Gong, A note on the size of the nilpotent residual in nite groups, Arch. Math. Vol. 99 (2012), pp. 413-416. DOI: 10.1007/s00013-012-0452-5 [2] Z. Halasi and K. Podoski, Bounds in groups with trivial Frattini subgroup, J. Algebra Vol. 319 (2008), pp. 893-896. DOI: 10.1016/j.jalgebra.2007.02.053 [3] P. Hall, The classication of prime power groups, J. Reine Angew. Math. Vol. 182 (1940), pp. 130-141. [4] P. Hall, The construction of soluble groups, Journal fr die reine und angewandte Mathematik Vol. 182 (1940), pp. 206-214, [5] N.S. Hekster, On the structure of n-isoclinism classes of groups, J. Pure Appl. Algebra Vol. 40 (1986), pp. 63-85. DOI.10.1016/0022-4049(86)90030-7 [6] M. Hezog, G. Kaplan and A. Lev, On the commutator and the center of nite groups, J. Algebra Vol. 278 (2004), pp. 494-501. DOI.10.1016/j.jalgebra.2004.03.021 [7] A.Y.E. Ol'shanskii, Varieties of nitely approximable groups, Izvestiya Rossiiskoi Akademii Nauk. Vol. 33 (1969), pp. 915-927. [8] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, (1982). [9] D.R. Taunt, On A-groups, Mathematical proceedings of the cambridge philosophical society Vol. 45 (1949), pp. 24-42. DOI: 10.1017/S0305004100000414 Published online: 24 October 2008 [10] J.H. Walter, The characterization of nite groups with abelian Sylow 2-subgroups, Annals of Mathematics Vol. 89 No. 3 (1969), pp. 405-514.