On the number of subgroups of given structure in a finite \(p\)-group
type: Item
instance of: scholarly article
title: On the number of subgroups of given structure in a finite \(p\)-group
zbMATH Open document ID: 0818.20017
DOI: 10.1007/BF01189883
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: As the title shows, the main interest of the author is to establish Anzahlsätze for finite \(p\)-groups. The main theorem of this paper will not be stated here as it needs too many preparatory definitions. Instead, to give a flavour of the results, I state only the last two ones. Let \(G\) be a finite \(p\)-group, let \(d(G)\) be the minimal number of generators of \(G\) and for \(1\leq i\leq d(G)\) let \(M_ i:= \{H< G\mid \Phi(G)\leq H\), \(| G:H|= p^ i\}\) and \(M_ i^ 0:=\{H\in M_ i\mid H\) is of maximal class\}. Proposition 11. If \(G\) is a 2-group and if \(G\) is not of maximal class, then for every \(i\) such that \(1\leq i\leq d(G)\) one has \(| M^ 0_ i |\equiv 0\pmod 4\). As a corollary, the following result of the author [Sib. Mat. Zh. 9, 1284-1306 (1968; Zbl 0315.20015)] is obtained: If a 2-group is not of maximal class, then the number of its subgroups of maximal class and of order at least 8 is a multiple of 4. Based on these facts, the author puts forward the following conjecture: If \(G\) is a group of order \(p^ m\), if \(1\leq i\leq d(G)\) and if \(i+ 1+p\leq m\), then \(| M^ 0_ i |\equiv 0\pmod {p^ 2}\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 687051
zbMATH Keywords: \(p\)-groups of maximal class
zbMATH Keywords: Anzahlsätze for finite \(p\)-groups
zbMATH Keywords: minimal number of generators
author: Yakov Berkovich
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
cites work: On the number of elements of given order in a finite \(p\)-group
cites work: Counting theorems for finite \(p\)-groups
cites work: On the Number of Solutions of the Equation x p k = a in a Finite p-Group
cites work: On the Number of Subgroups of Given Order in a Finite p-Group of Exponent p
cites work: On the Number of Subgroups of Given order and Exponent p in a Finite Irregular p -Group
cites work: On a special class of \(p\)-groups
cites work: Generalizations of Certain Elementary Theorems on p -Groups
cites work: Note on a paper of Berkovich
cites work: A Contribution to the Theory of Groups of Prime-Power Order
cites work: On a Theorem of Frobenius
cites work: Finite groups with small character degrees and large prime divisors
cites work: Realisierung von Darstellungen endlicher nilpotenter Gruppen. (Realization of representations of finite-dimensional nilpotent groups)
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description: scientific article; zbMATH DE number 687051