Commuting automorphisms of finite 2-groups of almost maximal class, II

Let [Formula: see text] be a finite [Formula: see text]-group. In our recent paper, it was shown that in a finite [Formula: see text]-group of almost maximal class, the set of all commuting automorphisms, [Formula: see text] is a subgroup of [Formula: see text]. Also, we proved that the minimum coclass of a non-[Formula: see text], [Formula: see text]-group is equal to 3. Using these results, in this paper, we will take of the task of determining when the group of all commuting automorphisms of all finite [Formula: see text]-groups of almost maximal class are equal to the group of all central automorphisms. This determination is not easy. We will prove they are equal, except only for five ones. We show that the minimum order of a [Formula: see text]-group which it’s group of all commuting automorphisms is not equal to it’s group of all central automorphisms is [Formula: see text]. Also, we prove that if [Formula: see text] is a finite [Formula: see text]-group in which [Formula: see text], then the subgroup of right 2-Engel elements of [Formula: see text], [Formula: see text], coincides with the second term of upper central series of [Formula: see text].

The absolute center of p-groups of maximal class

The purpose of this paper is to determine L(G), the absolute center of the group G, when G is a p-group of maximal class. Particularly we find L(G) for metabelian p-groups of maximal class, all p-groups of maximal class of order less than p6 and p-groups of maximal class for p = 2,3.

BOGOMOLOV MULTIPLIERS OF P-GROUPS OF MAXIMAL CLASS

Let $G$ be a $p$-group of maximal class and order $p^n$. We determine whether or not the Bogomolov multiplier ${\operatorname{B}}_0(G)$ is trivial in terms of the lower central series of $G$ and $P_1 = C_G(\gamma _2(G) / \gamma _4(G))$. If in addition $G$ has positive degree of commutativity and $P_1$ is metabelian, we show how understanding ${\operatorname{B}}_0(G)$ reduces to the simpler commutator structure of $P_1$. This result covers all $p$-groups of maximal class of large-enough order, and, furthermore, it allows us to give the first natural family of $p$-groups containing an abundance of groups with non-trivial Bogomolov multipliers. We also provide more general results on Bogomolov multipliers of $p$-groups of arbitrary coclass $r$.

Cleaning Data Sets with Diagnostic Errors in the High-Dimensional Feature Spaces

The paper proposes a new approach in data censoring, which allows correcting diagnostic errors in the data sets in case when these samples are described in high-dimensional feature spaces. Considering this case as a separate task is explained by the fact that in high-dimensional spaces most of the methods of outliers detection and data filtering, both statistical and metric, stop working. At the same time, for the tasks of medical diagnostics, given the complexity of the objects and phenomena studied, a large number of descriptive characteristics are the norm rather than the exception. To solve this problem, an approach that focuses on local similarity between objects belonging to the same class and uses the function of rival similarity (FRiS function) as a measure of similarity has been proposed. In this approach for efficient data cleaning from misclassified objects, the most informative and relevant low-dimensional feature subspace is selected, in which the separability of classes after their correction will be maximal. Class separability here means the similarity of objects of one class to each other and their dissimilarity to objects of another class. Cleaning data from class errors can consist both in their correction and removing the objects-outliers from the data set. The described method was implemented as a FRiS-LCFS algorithm (FRiS Local Censoring with Feature Selection) and tested on model and real biomedical problems, including the problem of diagnosing prostate cancer based on DNA microarray analysis. The developed algorithm showed its competitiveness in comparison with the standard methods for filtering data in high-dimensional spaces.