Linear Homogeneous Groups Research Articles

On sharp characters of rank 2 with rational values

For a finite group G and its character χ, let L be the set of distinct values of χ on non-identity elements of G and fL(χ(1)) denote the monic polynomial of least degree having L as its set of roots. Blichfeldt [A theorem concerning the invariants of linear homogeneous groups with some applications to substitution groups, Trans. Amer. Math. Soc. 5 (1904) 461–466. [2]] showed that fL(χ(1))/|G| is a rational integer. Cameron and Kiyota [Sharp character of finite groups, J. Algebra 115 (1988) 125–143] called the pair (G,χ)L-sharp if |G|=fL(χ(1)) and posed the problem of determining all the L-sharp pairs (G,χ) for a given set L. For several cases those problems have already been studied by many authors. Our purpose is to determine the sharp pairs of types {−2,1} and {−1,2} having non-trivial center.

Invariance theory for nonlocal variational principles—IV. Lagrangian classes

A Lagrangian coordinate or point class of a given finite continuous group consists of all Lagrangian functions that give action functionals that are invariant under the given group. Systems of linear or quasi linear first order partial differential equations are obtained whose general solution generate a Lagrangian class. The nonlocality is evidenced by the fact that the Lagrangian function and also the functions defining the functional arguments of the Lagrangian are the dependent variables in the system of partial differential equations. Complete integrability is shown when the group is a subgroup of the general linear homogeneous group and the dependent variables can be separated; the latter case is referred to as restricted. The paper concludes with a discussion of the invariance properties of the solutions of the nonlocal Euler equations.

The Linear Homogeneous Group, III

The Linear Homogeneous Group, III

The Degree of a Linear Homogeneous Group

The Degree of a Linear Homogeneous Group

The Linear Homogeneous Group. II

The Linear Homogeneous Group. II

The degree of a linear homogeneous group

The problem considered in this article is that of the relation between the degree of a linear homogeneous group and the abstract properties of the group. We have the theorem of Frobenius(l) to the effect that the degree of an irreducible group is a divisor of the order of the group, and then the generalization of this by Schur(2) which states that the order of the group is divisible by the product of the degree and the order of the central. In these Transactions(') I called attention to the importance of the commutators of the group in this connection by showing that the degree of an irreducible group is not less than the order of any invariant commutator. As a matter of fact, it is a multiple of this order. It is shown in the present paper that the orders of certain non-invariant commutators also have a bearing on the degree of a'linear group, whether it be reducible or irreducible. Moreover for groups of a certain category the degree is a multiple of the order of the whole commutator subgroup. If a linear homogeneous group has a similarity substitution as a commutator, the degree of the group is a multiple of the order of this commutator since all the multipliers of a similarity substitution are equal and the determinant of a commutator is equal to one. If G is an irreducible group of order g and degree n with a central of order a all its invariant substitutions are similarity substitutions, and if ,B is the order of an invariant commutator j32 is a divisor of g since g is divisible by na and a and n are both divisible by /. If G is reducible at least one of its irreducible components has an invariant commu'tator whose order is equal to the highest order of any multiplier of any invariant commutator of G. Therefore the degree of this component is a multiple of this order, and the degree of G is greater than this order. Hence we have the following theorem.

Generators of permutation groups simply isomorphic with 𝐿𝐹(2,𝑝ⁿ)

It is well known that the group LF(2, p) of linear fractional transformations of determinant unity in the GF[p] can be represented as a permutation group G of degree p+l. The purpose of this note is to show that the generators of G follow from a slight extension of an argument used in a recent paper. We obtain a representation of the abstract group L simply isomorphic with the special linear homogeneous group SLH(2, p) by means of the cosets K and KTS\, where X ranges over the p marks of the field ^o( = 0), #1, • • • , um, (m = p n — \). Let ko0 = K and kUi = KTSUi for i = 0, 1, • • • , m. If p is any mark, KSP = K and KTS\SP = KTS\+P, so that to Sp there corresponds the permutation

The Linear Homogeneous Group

This paper is a continuation of two other papers with the same title [1, 2]. Let GLn[R] be the group of n x n invertible matrices with elements from a ring R. In [1], R was the ring of integers modulo a prime power pr = v, and the normal subgroups of GLn[R] were described. For each ideal W in the ring R, two normal subgroups N*,n, Nvn were defined by means of congruences: Nvn {M det M = 1, M-I, identity, mod W}; N*,n = { M M mod W is scalar}. It was shown that every normal subgroup of GLn[R] (n>2) contains Nw n and is contained in N*,n for suitable W. This result was generalized in [8] to the case where R is any valuation ring. The important case where R is the ring of rational integers, and GLn[R] is the unimodular group, is not covered by either of these references. A study of the latter case is the primary purpose of this paper. The methods used here work reasonably well when R is any euclidean ring (even a non-commutative one); certain results hold in a still wider class of rings. The general problem of describing normal subgroups falls naturally into two parts. In the first part (?? 2, 3) the discussion is restricted to the case n > 2. There we show that if a normal subgroup N contains a certain matrix C, then N must contain also the canonical matrix A =I+ me2l for some positive integer m. The second part (?? 5, 6) includes the case n = 2. The problem here is to find the least normal subgroup Qmn of GLn[R], or of the subgroup SLn[R] of the latter, which contains A. For R = J, we conjecture that Qm,n is Nm,n = N(m),n, and establish the validity of this conjecture for m 2) into a form parallel to and generalizing the solution given in [1]; v need not be a prime power (? 4). The results of [8]

The invariants of forms under the binary linear homogeneous group 𝐺₆ Modulo 2

The invariants of forms under the binary linear homogeneous group 𝐺₆ Modulo 2

Invariants of the linear group modulo 𝑝^{𝑘}

Professor L. E. Dickson in his paper A fundamenttal system of invariants of the gener al miodular linear group wvith a solttion of the forvi problemt' gave a fundamental system of invariants for the group of all linear homogeneous transformations in n variables with coefficients in the Galois field of order pf, denoted by GF[pn]. Dr. J. S. Turner extended the results for the biniary group in GF[p] to the binary group H witlh coefficients ranging over integers modulo p2 and the determinant of transformation congruent to unity modulo p2, This thesis deals with the invariants of the linear homogeneous group in n variables with the coefficients ranging over integers modulo pk, and the determinant of transformation congruent to unity modulo p. The writer avails himself of this occasion to express his gratitude to Professor Dickson for the conduct of this research and for suggesting both the problem and the group-theoretic principles applied in the followinig pages.

On the Structure of Finite Continuous Groups

The structure of a finite continuous group is determined in part by the characteristic equation of the that is, the characteristic equation of the general infinitesimal transformation of the adjoint group of the given group. It is the purpose of this paper to show how certain properties of a group of linear homogeneous transformations can be obtained at once from the characteristic equation of the general infinitesimal transformation of the group, and thus how the type of structure is in part determined immediately from the infinitesimal transformations of such a group without the determination of the adjoint group. In ? 1 I show how to find the structural constants, and thus the roots of the characteristic equation, of the general infinitesimal transformation of the adjoint group of the general linear homogeneous group relative to a given subgroup; in ? 2 I find similar results for the special linear homogeneous group relative to a given subgroup. These results lead to certain relations, given in ? 3, between the characteristic equation of a group of linear homogeneous transformationcs and the characteristic equation of the general infinitesimal transformation of the given group. By the aid of these relations I establish theorems I-X by which certain properties of such a group can be at once determined from the general infinitesimal transformation of the group. Section 1.

On the order of linear homogeneous groups. IV

On the order of linear homogeneous groups. IV

On the Order of Linear Homogeneous Groups: (Fourth Paper)

On the Order of Linear Homogeneous Groups: (Fourth Paper)

Concerning the degree of an irreducible linear homogeneous group

Concerning the degree of an irreducible linear homogeneous group

Irreducible Linear Homogeneous Groups Whose Orders Are Powers of a Prime

Irreducible Linear Homogeneous Groups Whose Orders Are Powers of a Prime

Irreducible linear homogeneous groups whose orders are powers of a prime

The purpose of this article is to determine as far as possible the connection between the degree of an irreducible linear homogeneous group and its abstract group properties. The discussion will be limited for the mnost part to groups whose orders are powers of a prime. As a particular phase of the general subject there arises the question as to the circumstances under which a group can be simply isomorphic with irreducible groups of different degrees. The simple group of order 168, for example, is simply isomorphic with irreducible groups of degrees 3, 6, 7, and 8 respectively. On the other hand, I know of no group whose order is a power of a prime that is simply isomorphic with irreducible groups of different degrees. In fact, the following discussion shows that in certain irreducible groups the degree is uniquiely fixed by certain abstract properties of the group; and if the degree is not thus uniquely fixed the order of the group must be greater than the seventh power of a prime. THEOREM I. A linear honmogeneous group G, of classt k, all of whose invariant operations are similarity substitutions either is irreducible or is simply isomorphic with each of its irreducible components. lf G is reducible, suppose that it has been put into its completely reduced formii. The invariant operations will niot be affected by this change. If in alny operation the identity of any irreducible component were associated with a nonidentical operation in the other variables, this operation would be non-invariant in G., and would therefore give at least one non-identical commutator. This comnmutator would also be non-invariant, and none of its successive commutators, except identity, could be invariant. But this is impossible since G is of class k. Hence every irreducible component of G is simply isomorphic with G. THEOREM II. A linear honTogeneous group G, qf order a power of a prime, that has a cyclic central either is irreducible or is simply isomorphic with at leaist one of its irreducible components.

On the Order of Linear Homogeneous Groups (Supplement)

On the Order of Linear Homogeneous Groups (Supplement)

On the order of linear homogeneous groups (supplement)

On the order of linear homogeneous groups (supplement)

On the Quaternary Linear Homogeneous Groups Modulo p of Order a Multiple of p

On the Quaternary Linear Homogeneous Groups Modulo p of Order a Multiple of p

Subgroups of Order a Power of p in the General and Special m-ary Linear Homogeneous Groups in the GF[p n

1. It would seem that the most effective method of determining all the subgroups of order a multiple of p of a l'inear group in the Galois field of order pn is that based upon a complete knowledge of the subgroups of order a power of p. This method has proved successful for the ternary groupsl and, as I will show on another occasion, also for the quaternary groups. The present investigation proceeds far enough to give a clear insight into the nature of the simple laws pervading the subject. It is hoped that the results are capable of extension by induction to all powers of p. To indicate the difficulty of this step, it may be remarked that its completion would give the means of deriving at once an explicit list of all groups of order a power of a prime, and simultaneously all the subgroups of each. A second aim of the paper was to furnish data for the problem of the determination of all m-ary groups for low values of m. Following Lie, I write (A, B) for the commutator A-B1AB. I employ the usual notation Bj for the transformation which alters only i, replacing it by 6; + afj. We have the simple relation ?