scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

On the Minimum Order of Graphs with Given Group

1974 ◽  
Vol 17 (4) ◽  
pp. 467-470 ◽  
Author(s):  
László Babai
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Minimum Order ◽  
Number Of Generators ◽  
Minimum Number ◽  
A Finite Group

For G a, finite group let α(G) denote the minimum number of vertices of the graphs X the automorphism group A(X) of which is isomorphic to G.G. Sabidussi proved [1], that α(G)=0(n log d) where n=\G\ and d is the minimum number of generators of G.As 0(log n) is the best possible upper bound for d, the result established in [1] implies that α(G)=0(n log log n).

scholarly journals Linear groups analogous to permutation groups

1963 ◽  
Vol 3 (2) ◽  
pp. 180-184 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Finite Linear Group ◽  
Singular Matrices ◽  
A Finite Group ◽  
Complex Coefficients ◽  
Finite Linear

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

scholarly journals The Chromatic Number of Finite Group Cayley Tables

10.37236/7874 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Kevin Halasz ◽  
E. S. Mahmoodian
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Chromatic Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Latin Square ◽  
Finite Abelian Groups ◽  
Cayley Table ◽  
Minimum Number ◽  
Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

Finite Linear Groups of Degree Seven. I

1969 ◽  
Vol 21 ◽  
pp. 1042-1053 ◽  
Author(s):  
David B. Wales
Keyword(s):  
Finite Group ◽  
Complex Representation ◽  
Linear Groups ◽  
Subsequent Paper ◽  
Prime Degree ◽  
Sylow Group ◽  
A Finite Group ◽  
Finite Linear

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

scholarly journals On the odd order composition factors of finite linear groups

2020 ◽  
Vol 23 (6) ◽  
pp. 1057-1068
Author(s):  
Alexander Betz ◽  
Max Chao-Haft ◽  
Ting Gong ◽  
Anthony Ter-Saakov ◽  
Yong Yang
Keyword(s):  
Linear Group ◽  
Linear Groups ◽  
Composition Series ◽  
Odd Order ◽  
Finite Linear Group ◽  
Composition Factors ◽  
Finite Linear

AbstractIn this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group. First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order. Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.

Polynomial Invariants of Finite Linear Groups of Degree Two

1980 ◽  
Vol 32 (2) ◽  
pp. 317-330 ◽  
Author(s):  
W. Cary Huffman
Keyword(s):  
Finite Group ◽  
Invariant Theory ◽  
Weight Enumerator ◽  
Linear Groups ◽  
Weight Enumerators ◽  
Polynomial Invariants ◽  
Matrix Groups ◽  
A Finite Group ◽  
Finite Linear

Recently invariant theory of linear groups has been used to determine the structure of several weight enumerators of codes. Under certain conditions on the code, the weight enumerator is invariant under a finite group of matrices. Once all the polynomial invariants of this group are known, the form of the weight enumerator is restricted and often useful results about the existence and structure of codes can be found. (See [5], [8], [14], and [15].) Many of the groups in these applications are of degree 2; in this paper all the invariants of finite 2 X 2 matrix groups over C are determined.

Finite Linear Groups of Prime Degree

1969 ◽  
Vol 21 ◽  
pp. 1025-1041 ◽  
Author(s):  
David B. Wales
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Irreducible Representations ◽  
Complex Representation ◽  
Linear Groups ◽  
Prime Degree ◽  
Long Time ◽  
A Finite Group ◽  
Finite Linear

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.

scholarly journals Wirtinger numbers for virtual links

2019 ◽  
Vol 28 (14) ◽  
pp. 1950086
Author(s):  
Puttipong Pongtanapaisan
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knots ◽  
Link Group ◽  
Number Of Generators ◽  
Virtual Links ◽  
Minimum Number ◽  
Bridge Number

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh’s Tube map to these knots we can obtain nontrivial weakly superslice links.

scholarly journals Growth sequences of finite groups V

1981 ◽  
Vol 31 (3) ◽  
pp. 374-375 ◽  
Author(s):  
D. Meier ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
The Other ◽  
Direct Power ◽  
Easy Proof ◽  
Number Of Generators ◽  
Other Hand ◽  
Minimum Number

AbstractA short and easy proof that the minimum number of generators of the nth direct power of a non-trival finite group of order s having automorphism group of order a is more than logsn + logsa, n > 1. On the other hand, for non-abelian simple G and large n, d(Gn) is within 1 + e of logsn + logsa.

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