On the converse of Baer's theorem for generalizations of groups with trivial Frattini subgroups

Document Type : Research Paper

Authors

Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran

Abstract

In 2012, Guo and Gong proved that if $G$ is a finite nonabelian group with $\Phi(G)=1$, then $|G:Z(G)|<|G'||U(G)|$, in which $U(G)$ is the nilpotent residual of $G$. We show that the assumption of finiteness of the group can be omitted. Moreover, we investigate converse of Schur and Baer's theorems for groups that can be seen as  generalizations of groups with trivial Frattini subgroups  and state some properties of $n$-isoclinism families  of these groups.

Keywords

References

[1] S. Dol , M. Herzog, G. Kaplan and A. Lev, On the size of the nilpotent residual in nite groups, J. Group
Theory Vol. 10 (2007), pp. 299-305.
[2] 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.
[3] Z. Halasi and K. Podoski, Bound in groups with trivial Frattini subgroup, J. Algebra Vol. 319 No. 3 (2008)
pp. 893-896.
[4] P. Hall, Finite-by-nilpotent groups, Proc. Camb. Phil. Soc. Vol. 52 (1956), pp. 611-616.
[5] R. Hatamian, M. Hassanzadeh and S. Kayvanfar, A converse of Baer's theorem, Ric. mat. Vol. 63 (2014),
pp. 183-187.
[6] N. S. Hekster, On the structure of n-isoclinism classes of groups, J. Pure Appl. Algebra Vol. 40 No. 1(1986),
pp. 63-85.
[7] N. S. Hekster, Varieties of groups and isologisms, J. Austral. Math. Soc. (Series A) Vol. 46 (1989), pp.
22-60.
[8] M. Herzog, G. Kaplan and A. Lev, On the commutator and the center of nite groups, J. Algebra Vol. 728
(2004), pp. 494-501.
[9] M. Herzog, G. Kaplan and A. Lev, The size of the commutator subgroup of nite groups, J. Algebra Vol.
320 (2008), pp. 980-986.
[10] L. C. Kappe and J. Kirtland, Finite group with trivial Frattini subgroups, Arch. Math. Vol. 80 (2003), pp.
225-234.
[11] K. Podoski and B. Szegedy, Bounds for the index of the centre in capable groups, Proc. Amer. Math. Soc.
Vol. 133 No. 12 (2005), pp. 3441-3445.
[12] D. J. S. Robinson, A Course in the Theory of Groups, Springer, Berlin, (1996).
Algebraic Structures and Their Applications
Volume 6, Issue 1
March 2019
Pages 139-148

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