Inducing characters and nilpotent subgroups

If is a subgroup of a finite group and induces irreducibly up to , we prove that, under certain odd hypothesis, is a nilpotent subgroup of .

     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

Fuzzy subgroups of nilpotent groups

In 1991, Asaad has suggested the following question: What can be said about the structure of a group G when information is known about the structure of a fuzzy subgroup A of G and vice-versa? This question has been partially investigated in Fuzzy Sets and Systems39 (1991) 323–328. The purpose of this paper is to continue more investigations for the above mentioned question.     

Conjugacy classes of maximal nilpotent subgroups

It is shown that in an arbitrary finite groupG, any two maximal nilpotent subgroups ofG whose intersection contains its own centralizer inG, are conjugate inG.     

On the existence of stably nilpotent subgroups in locally nilpotent groups

A condition for the existence of stably nilpotent subgroups of a given nilpotency class in a locally nilpotent group is established.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 361–368, September, 1968.     

On subgroups of residually nilpotent metabelian groups

ABSTRACT

We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006 Faith , C. ( 2006 ). Factor rings of pseudo-Frobenius rings . J. Algebra and Its Applications 6 :(to appear). [CSA] [Web of Science ®] , [Google Scholar]). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5).

These results also generalize results of the author in Faith (1985 Faith , C. ( 1985 ). The maximal regular ideal of self-injective and continuous rings splits off . Arch. Math. 44 : 511 – 521 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring.

The results are applied in Section 6 to extend theorems of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) characterizing VNR rings, and, as the title of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) suggests, extend the conjecture of Shamsuddin.     

Large nilpotent subgroups of finite simple groups

Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|²<|G|. Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000.     

Finite groups generated by two nilpotent subgroups

Some criteria for the solvability of ABA groups with nilpotent subgroups A and B of special form are proved. An ABA group is a group of the form G = ABA, where A and B are subgroups of the group G.Translated from Matematicheskie Zametki, Vol. 11, No. 3, pp. 293–298, March, 1972.     

Primitive characters of subgroups ofM-groups

Research partially supported by DGICYT.PB 90-0414-C03-01     

Finite solvable groups with nilpotent 2-maximal subgroups

In this article we describe finite solvable groups whose 2-maximal subgroups are nilpotent (a 2-maximal subgroup of a group). Unsolvable groups with this property were described in [2,3].Translated from Matematicheskie Zamestki, Vol. 3, No. 1, pp. 21–32, January, 1968.