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.

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.

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.

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 SPECTRUM OF LINEAR GROUPS OVER THE BINARY FIELD AND RECOGNIZABILITY OF L12(2)

2006 ◽  
Vol 16 (02) ◽  
pp. 341-349 ◽  
Author(s):  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Computer Program ◽  
Simple Group ◽  
Linear Groups ◽  
Element Orders ◽  
Binary Field ◽  
Set Of Element Orders ◽  
A Finite Group

The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable through its spectrum, if for every finite group H, the equality of the spectra ω(H) = ω(G) implies the isomorphism H ≅ G. In this paper, first we try to write a computer program for computing ω(Ln(2)) for any n ≥ 3. Then, we will show that the simple group L12(2) is recognizable through its spectrum.

Recognition of Finite Simple Linear Groups L4(2k) by Spectrum

2010 ◽  
Vol 17 (03) ◽  
pp. 469-474
Author(s):  
Mingchun Xu
Keyword(s):  
Finite Group ◽  
Linear Groups ◽  
Element Orders ◽  
Set Of Element Orders ◽  
A Finite Group

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. Grechkoseeva, Shi and Vasilev have proved that the simple linear groups Ln(2k) are recognizable by spectrum for n=2m≥ 16. In this paper we establish the recognizability for the case n=4.

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.

A NEW CHARACTERIZATION OF SOME LINEAR GROUPS BY NSE

2013 ◽  
Vol 13 (02) ◽  
pp. 1350094 ◽  
Author(s):  
CHANGGUO SHAO ◽  
QINHUI JIANG
Keyword(s):  
Finite Group ◽  
Linear Group ◽  
Linear Groups ◽  
Element Orders ◽  
Set Of Element Orders ◽  
A Finite Group

Let G be a finite group and πe(G) be the set of element orders of G. Assume that k ∈ πe(G) and mk(G) is the number of elements of order k in G. Set nse (G) ≔ {mk(G) | k ∈ πe(G)}, we call nse (G) the set of numbers of elements with same order. In this paper, we give a new characterization of simple linear group L2(2a) by its order |L2(2a)| and the set nse (L2(2a)), where either 2a - 1 or 2a + 1 is a prime.

scholarly journals On isomorphisms of connected Cayley graphs, III

1998 ◽  
Vol 58 (1) ◽  
pp. 137-145 ◽  
Author(s):  
Cai Heng Li
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Simple Group ◽  
Prime Divisor ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Linear Groups ◽  
Finite Simple Groups ◽  
Natural Question ◽  
A Finite Group

For a finite group G and a subset S of G which does not contain the identity of G, we use Cay(G, S) to denote the Cayley graph of G with respect to S. For a positive integer m, the group G is called a (connected) m-DCI-group if for any (connected) Cayley graphs Cay(G, S) and Cay(G, T) of out-valency at most m, Sσ = T for some σ ∈ Aut(G) whenever Cay(G, S) ≅ Cay(G, T). Let p(G) be the smallest prime divisor of |G|. It was previously shown that each finite group G is a connected m-DCI-group for m ≤ p(G) − 1 but this is not necessarily true for m = p(G). This leads to a natural question: which groups G are connected p(G)-DCI-groups? Here we conjecture that the answer of this question is positive for finite simple groups, that is, finite simple groups are all connected 2-DCI-groups. We verify this conjecture for the linear groups PSL(2, q). Then we prove that a nonabelian simple group G is a 2-DCI-group if and only if G = A5.

scholarly journals HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS

2015 ◽  
Vol 16 (2) ◽  
pp. 351-419 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Paul Baum ◽  
Roger Plymen ◽  
Maarten Solleveld
Keyword(s):  
Finite Group ◽  
Local Field ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Linear Groups ◽  
Special Linear Groups ◽  
Affine Hecke Algebra ◽  
Morita Equivalent ◽  
Affine Hecke Algebras ◽  
A Finite Group

Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

RECOGNIZING BY ORDER AND DEGREE PATTERN OF SOME PROJECTIVE SPECIAL LINEAR GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250051 ◽  
Author(s):  
B. AKBARI ◽  
A. R. MOGHADDAMFAR
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Linear Groups ◽  
Special Linear ◽  
Projective Special Linear Groups ◽  
Special Linear Groups ◽  
Isomorphism Classes ◽  
Degree Pattern ◽  
A Finite Group

Let M be a finite group and D (M) be the degree pattern of M. Denote by h OD (M) the number of isomorphism classes of finite groups G with the same order and degree pattern as M. A finite group M is called k-fold OD-characterizable if h OD (M) = k. Usually, a 1-fold OD-characterizable group is simply called OD-characterizable. The purpose of this article is twofold. First, it provides some information on the structure of a group from its degree pattern. Second, it proves that the projective special linear groups L4(4), L4(8), L4(9), L4(11), L4(13), L4(16), L4(17) are OD-characterizable.

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