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.     

Large abelian normal subgroups

In this paper we study the family of finite groups with the property that every maximal abelian normal subgroup is self-centralizing. It is well known that this family contains all finite supersolvable groups, but it also contains many other groups. In fact, every finite group G is a subgroup of some member \(\Gamma \) of this family, and we show that if G is solvable, then \(\Gamma \) can be chosen so that every abelian normal subgroup of G is contained in some self-centralizing abelian normal subgroup of \(\Gamma \).     

Groups with maximal subgroups of Sylow subgroups normal

This paper characterizes those finite groups with the property that maximal subgroups of Sylow subgroups are normal. They are all certain extensions of nilpotent groups by cyclic groups.     

Character degree graphs and normal subgroups

We consider the degrees of those irreducible characters of a group whose kernels do not contain a given normal subgroup . We show that if and is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of and of .

     

Groups with dense normal subgroups

A group is said to have dense normal subgroups, if each non-empty open interval in its lattice of subgroups contains a normal subgroup. The structure of this and related classes of groups is investigated. Typical results are: an infinite group with dense ascendant subgroups is locally nilpotent: a nontorsion group with dense normal subgroups is abelian, etc.     

p-groups without noncharacteristic normal subgroups

Let P be a finite p-group, p a prime. We prove that there is a finite p-group Q ≥ P such that every normal subgroup of Q is characteristic in Q.     

On distortion of normal subgroups

Geometriae Dedicata - We examine distortion of finitely generated normal subgroups. We show a connection between subgroup distortion and group divergence. We suggest a method computing the...