scholarly journals On Abelian Permutation Groups

1965 ◽  
Vol 8 (5) ◽  
pp. 627-630 ◽  
Author(s):  
R. Bercov ◽  
L. Moser
Keyword(s):  
Symmetric Group ◽  
Abelian Subgroup ◽  
Maximal Order ◽  
Permutation Groups ◽  
Abelian Subgroups ◽  
Image Position

The principal object of this note is to determine the maximal order of Abelian subgroups of the symmetric group sn of degree n. We also discuss some related results and problems.A largest Abelian subgroup of sn has order f(n) where

Maximal Abelian Subgroups of the Symmetric Groups

1971 ◽  
Vol 23 (3) ◽  
pp. 426-438 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Symmetric Group ◽  
General Problem ◽  
Abelian Subgroup ◽  
Symmetric Groups ◽  
Specific Class ◽  
Maximal Abelian Subgroup ◽  
Number Of Generators ◽  
Abelian Subgroups ◽  
Almost All

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

scholarly journals On maximal abelian groups of maps

1993 ◽  
Vol 55 (3) ◽  
pp. 414-420 ◽  
Author(s):  
Reinhard Winkler
Keyword(s):  
Symmetric Group ◽  
Fixed Points ◽  
Abelian Subgroup ◽  
Abelian Groups ◽  
Simple Description ◽  
Abelian Subgroups

AbstractThe paper gives a rather simple description of all maximal abelian subgroups H of the symmetric group SM acting on an arbitrary set M. In the case of finite M this result is used to determine the maximal cardinality of such an H and the maximal number of permutations without fixed points contained in an abelian subgroup of SM.

scholarly journals Large abelian subgroups of Chevalley groups

1979 ◽  
Vol 27 (1) ◽  
pp. 59-87 ◽  
Author(s):  
Michael J. J. Barry
Keyword(s):  
Finite Field ◽  
Chevalley Group ◽  
Abelian Subgroup ◽  
Maximal Order ◽  
Chevalley Groups ◽  
Abelian Subgroups

AbstractFor any group S let Ab(S) = {A∣A is an abelian subgroup of S of maximal order}. Let G be a Chevalley group of type An, Bn, Cn, or Dn over a finite field of characteristic p and let. In this paper Ab(U) is determined for all such groups.

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

scholarly journals On infinite groups in which all abelian subgroups are locally cyclic

2021 ◽  
Author(s):  
Costantino Delizia ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Periodic Case ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Periodic Groups ◽  
Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

Frobenius Symbols and the Groups SsGL(n), O(n) and Sp(n)

1973 ◽  
Vol 25 (5) ◽  
pp. 941-959 ◽  
Author(s):  
Y. J. Abramsky ◽  
H. A. Jahn ◽  
R. C. King
Keyword(s):  
Symmetric Group ◽  
Irreducible Representations ◽  
Image Position

Frobenius [2; 3] introduced the symbolsto specify partitions and the corresponding irreducible representations of the symmetric group Ss.

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

GROUPS SATISFYING THE DOUBLE CHAIN CONDITION ON ABELIAN SUBGROUPS

2018 ◽  
Vol 99 (2) ◽  
pp. 212-218 ◽  
Author(s):  
MATTIA BRESCIA ◽  
ALESSIO RUSSO
Keyword(s):  
Chain Condition ◽  
Double Sequences ◽  
Property A ◽  
Double Chain ◽  
Abelian Subgroups ◽  
Image Position

If $\unicode[STIX]{x1D703}$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $\unicode[STIX]{x1D703}$-subgroups if it admits no infinite double sequences $$\begin{eqnarray}\cdots <X_{-n}<\cdots <X_{-1}<X_{0}<X_{1}<\cdots <X_{n}<\cdots\end{eqnarray}$$ consisting of $\unicode[STIX]{x1D703}$-subgroups. We describe the structure of generalised radical groups satisfying the double chain condition on abelian subgroups.

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