Character stabilizer limits relative to a normal nilpotent subgroup

Relative non-Normal Graphs of a Subgroup of Finite Groups

Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...

Normal subgroup growth in free class-2-nilpotent groups

Let F2,d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions ζ F2,d(s), prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

Automorphisms Fixing Every Normal Subgroup of a Nilpotent-by-abelian Group

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class c)-by-abelian group is (nilpotent of class ≤ c)-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most 3. An example shows that this bound cannot be improved.

Estimating a Bounded Normal Mean Relative to Squared Error Loss Function

Let be a random sample from a normal distribution with unknown mean and known variance The usual estimator of the mean, i.e., sample mean is the maximum likelihood estimator which under squared error loss function is minimax and admissible estimator. In many practical situations, is known in advance to lie in an interval, say for some In this case, the maximum likelihood estimator...