On subgroups containing non-trivial normal subgroups

We prove that ifA≠1 is a subgroup of a finite groupG and the order of an element in the centralizer ofA inG is strictly larger (larger or equal) than the index [G:A], thenA contains a non-trivial characteristic (normal) subgroup ofG. Consequently, ifA is a stabilizer in a transitive permutation group of degreem>1, thenexp(Z(A))<m. These theorems generalize some recent results of Isaacs and the authors.     

On commutator subgroups

Acta Mathematica Hungarica -     

On the permutability of sylow subgroups with Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Sufficient conditions are established for the p-solvability of a finite group in which a Sylow p-subgroup is permutable with some Schmidt subgroups. Sufficient conditions for the solvability of a finite group in which some Schmidt subgroups are permutable are also obtained.     

On second maximal subgroups of Sylow subgroups of finite groups

Finite groups in which the second maximal subgroups of the Sylow p-subgroups, p a fixed prime, cover or avoid the chief factors of some of its chief series are completely classified.     

On 2-isolated subgroups

One characteristic property of ZT-groups is proved.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 497–502, May, 1968.     

On invariant additive subgroups

Suppose thatR is a prime ring with the centerZ and the extended centroidC. An additive subgroupA ofR is said to be invariant under special automorphisms if (1+t)A(1+t)⁻¹ ⊆A for allt ∈R such thatt ²=0. Assume thatR possesses nontrivial idempotents. We prove: (1) If chR ≠ 2 or ifRC ≠C ₂, then any noncentral additive subgroup ofR invariant under special automorphisms contains a noncentral Lie ideal. (2) If chR=2,RC=C ₂ andC ≠ {0, 1}, then the following two conditions are equivalent: (i) any noncentral additive subgroup invariant under special automorphisms contains a noncentral Lie ideal; (ii) there isα ∈Z / {0} such thatα ² Z ⊆ {β ²:β ∈Z}.     

On the n-minimal subgroups

Let G be a finite group and n be a positive integer. An n-minimal subgroup H of G is called to be exactly n-minimal if no proper subgroup of H is n-minimal. In this paper, we study the solvability of G under the assumption that all exactly n-minimal subgroups of G are S-permutable.     

On R-conjugate-permutability of sylow subgroups

A subgroup H of a finite group G is said to be conjugate-permutable if HH^g = H^gH for all g ∈ G. More generaly, if we limit the element g to a subgroup R of G, then we say that the subgroup H is R-conjugate-permutable. By means of the R-conjugatepermutable subgroups, we investigate the relationship between the nilpotence of G and the R-conjugate-permutability of the Sylow subgroups of A and B under the condition that G = AB, where A and B are subgroups of G. Some results known in the literature are improved and generalized in the paper.