scholarly journals KAZHDAN CONSTANTS OF GROUP EXTENSIONS

2010 ◽  
Vol 20 (05) ◽  
pp. 671-688
Author(s):  
UZY HADAD
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Group Extensions ◽  
Abelian Extensions ◽  
Tame Automorphism ◽  
Relatively Free Group

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of p-groups of lower p-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.

scholarly journals The skeleton of a variety of groups

1972 ◽  
Vol 6 (3) ◽  
pp. 357-378 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Product Variety ◽  
Finite Variety ◽  
Locally Finite ◽  
Variety Of Groups ◽  
Locally Finite Variety ◽  
Countably Infinite ◽  
Relatively Free Group

The skeleton of a variety of groups is defined to be the intersection of the section closed classes of groups which generate . If m is an integer, m > 1, is the variety of all abelian groups of exponent dividing m, and , is any locally finite variety, it is shown that the skeleton of the product variety is the section closure of the class of finite monolithic groups in . In particular, S) generates . The elements of S are described more explicitly and as a consequence it is shown that S consists of all finite groups in if and only if m is a power of some prime p and the centre of the countably infinite relatively free group of , is a p–group.

Automorphisms of free nilpotent-by-abelian groups

1993 ◽  
Vol 114 (1) ◽  
pp. 143-147 ◽  
Author(s):  
R. M. Bryant ◽  
C. K. Gupta
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Metabelian Groups ◽  
Verbal Subgroup ◽  
Variety Of Groups ◽  
Relatively Free Group

Let Fn be a free group of finite rank n with basis {x1,…, xn}. Let be a variety of groups and write for the verbal subgroup of Fn corresponding to . (See [11] for information on varieties and related concepts.) Every automorphism of Fn induces an automorphism of the relatively free group Fn/V, and those automorphisms of Fn/V arising in this way are called tame. If is the variety of all metabelian groups and n ╪ 3 then every automorphism of Fn/V is tame [2, 4, 12]. But this is an exceptional situation. For many (and probably most) other varieties , Fn/V has non-tame automorphisms for all sufficiently large n. This holds for the variety of all nilpotent groups of class at most c where c ≥ 3 [1, 3] and for nearly all product varieties including, in particular, the variety of all groups whose derived groups are nilpotent of class at most c, where c > 2 [10, 13].

Pointwise bound for ℓ-torsion in class groups: Elementary abelian extensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiuya Wang
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Abelian Groups ◽  
Class Groups ◽  
Abelian Extensions ◽  
Pointwise Bound ◽  
First Case

AbstractElementary abelian groups are finite groups in the form of {A=(\mathbb{Z}/p\mathbb{Z})^{r}} for a prime number p. For every integer {\ell>1} and {r>1}, we prove a non-trivial upper bound on the {\ell}-torsion in class groups of every A-extension. Our results are pointwise and unconditional. This establishes the first case where for some Galois group G, the {\ell}-torsion in class groups are bounded non-trivially for every G-extension and every integer {\ell>1}. When r is large enough, the unconditional pointwise bound we obtain also breaks the previously best known bound shown by Ellenberg and Venkatesh under GRH.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

scholarly journals Automorphisms of nilpotent groups of class two with small rank

1985 ◽  
Vol 39 (1) ◽  
pp. 121-131 ◽  
Author(s):  
Thomas A. Fournelle
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Direct Sums ◽  
Matrix Computations ◽  
Central Factor ◽  
Rank One ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractRational abelian groups, that is, torsion-free abelian groups of rank one, are characterized by their types. This paper characterizes rational nilpotent groups of class two, that is, nilpotent groups of class two in which the center and central factor group are direct sums of rational abelian groups. This characterization is done according to the types of the summands of the center and the central factor group. Using these types and some cohomological techniques it is possible to determine the automorphism group of the nilpotent group in question by performing essentially matrix computations.In particular, the automorphism groups of rational nilpotent groups of class two and rank three are completely described. Specific examples are given of semicomplete and pseudocomplete nilpotent groups.

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

Finite Groups with Automorphism Groups Having Orders 4pq2 (2 < p < q) II

2017 ◽  
Vol 24 (01) ◽  
pp. 153-168
Author(s):  
Yu Zeng ◽  
Jinbao Li ◽  
Guiyun Chen
Keyword(s):  
Finite Groups ◽  
Automorphism Groups ◽  
Nilpotent Groups ◽  
The Third

The authors study and classify finite groups with their automorphism groups having orders 4pq2, where p and q are primes such that 2 < p < q. In 2013 the first and the third authors classified the nilpotent groups of this kind; here the authors give the classification of those finite non-nilpotent groups with their automorphism groups having orders 4pq2.

Representations of holomorphs of group extensions with Abelian kernels

1975 ◽  
Vol 78 (3) ◽  
pp. 357-368 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Finite Rank ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Group Extensions

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

scholarly journals Conformality and p-isomorphism in finite nilpotent groups

1967 ◽  
Vol 7 (2) ◽  
pp. 165-171 ◽  
Author(s):  
C. D. H. Cooper
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Equivalence Relations ◽  
The Relationship

This paper discusses the relationship between two equivalence relations on the class of finite nilpotent groups. Two finite groups are conformal if they have the same number of elements of all orders. (Notation: G ≈ H.) This relation is discussed in [4] pp 107–109 where it is shown that conformality does not necessarily imply isomorphism, even if one of the groups is abelian. However, if both groups are abelian the position is much simpler. Finite conformal abelian groups are isomorphic.

MINIMAL NON-ABELIAN GROUPS AS AUTOMORPHISM GROUPS OF FINITE GROUPS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350150
Author(s):  
S. FOULADI ◽  
R. ORFI
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Abelian Groups

In this paper, we classify all finite groups whose automorphism group is minimal non-abelian.

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