Classes d'éléments conjugués des groupes cristallographiques

1978 ◽  
Vol 34 (6) ◽  
pp. 895-900
Author(s):  
J. Sivardière
Keyword(s):  
Finite Group ◽  
Double Point ◽  
Invariant Subgroup ◽  
Factor Group ◽  
Direct Products ◽  
Simple Point ◽  
Space Groups ◽  
Point Groups ◽  
A Finite Group

Let G be a finite group, H an invariant subgroup and F the corresponding factor group. The classes of conjugated elements of G are derived from the classes of H and F. We consider simple point groups and symmorphic space groups, which are semi-direct products H^F, then double point groups and non- symmorphic space groups, which are extensions of F by H.

The crystallographic point groups as semi-direct products

1963 ◽  
Vol 255 (1054) ◽  
pp. 216-240 ◽  
Keyword(s):  
Direct Product ◽  
Invariant Subgroup ◽  
Projection Operators ◽  
Direct Products ◽  
Finite Rotations ◽  
Space Groups ◽  
Symmetry Adapted Functions ◽  
Point Groups ◽  
Definition Of

This paper aims at providing a systematic treatment of the crystallographic point groups. Some well-known properties of them, in terms of the theory of the poles of finite rotations, are first discussed, so as to provide a simple way for recognizing their invariant subgroups. A definition of the semi-direct product is then given, and it is shown that all crystallographic point groups can be expressed as a semi-direct product of one of their invariant subgroups by a cyclic subgroup. Many useful relations between point groups can be obtained by exploiting the properties of the triple and mixed triple semi-direct products, which are defined. Much of the rest of the paper is devoted to the theory of the representations of semi-direct products. The treatment here parallels that given by Seitz (1936) for the reduction of space groups in terms of the representations of its invariant subgroups (the translation groups). The latter, however, are always Abelian and this is not always the case for point groups. The full treatment of the general case, such as given by McIntosh (1958), is laborious and it is shown that, if the emphasis is placed on the bases of the representations, rather than the representations themselves, it is possible to achieve the reduction of the point groups by a method hardly more involved than that required when the invariant subgroup is Abelian. It is also shown that, just as for space groups, the representations of the invariant subgroups can be denoted and visualized by means of a vector, which allows a very rapid classification of the representations, very much as the k vector as used by Bouckaert, Smoluchowski & Wigner (1936) allows the formalism of the Seitz method for space groups to be carried out in a graphical fashion. One of the major consequences of this work is that it affords a substantial simplification in the use of the symmetrizing and projection operators that are required to obtain symmetry-adapted functions: a very systematic alternative to the method given by Melvin (1956) is therefore provided. In the last section of the paper all the techniques discussed are applied in detail, as an example, to the cubic groups. The projection operators are used to obtain symmetry-adapted spherical harmonics for these groups. The paper might be found useful as an introduction to the methods for the reduction of space groups.

Ray representations of point groups and the irreducible representations of space groups and double space groups

1966 ◽  
Vol 260 (1108) ◽  
pp. 1-36 ◽  
Keyword(s):  
Finite Group ◽  
Irreducible Representations ◽  
High Symmetry ◽  
Full Matrix ◽  
Space Groups ◽  
Double Space ◽  
Point Groups ◽  
A Finite Group ◽  
Representations Of Space ◽  
Complex Cases

The theory of the ray representations of a finite group is summarized and full matrix ray representations are derived and tabulated for all thirty-two point groups. It is shown that any irreducible representation of any of the 230 space groups and of the corresponding double groups may be obtained quickly and easily from these ray representations of the point groups. The most complex cases which arise, namely points of high symmetry on the surface of the Brillouin zone for the regular holohedric space groups, 01 ... OJ°, are treated explicitly. The relation of the present work to the recent treatments of Slater and Kovalev is discussed.

Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts

2015 ◽  
Vol 230 (12) ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Kushan Ratnayake ◽  
Sajdeh Sajjadi
Keyword(s):  
Direct Product ◽  
Free Space ◽  
Special Class ◽  
The Other ◽  
Pure Point ◽  
Direct Products ◽  
Space Groups ◽  
Point Transformations ◽  
Point Groups ◽  
Torsion Free

AbstractPoint groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct product of pure translations and pure point subgroups. In contrast, Bieberbach groups consist of pure translations, screws, and glides. These “torsion-free” space groups are rarely mentioned as being a special class outside of the mathematics literature. Every space group can be thought of as lying along a spectrum with the symmorphic case at one extreme and Bieberbach space groups at the other. The remaining nonsymmorphic space groups lie somewhere in between. Many of these can be decomposed into semi-direct products of Bieberbach subgroups and point transformations. In particular, we show that those 3D Sohncke space groups most populated by macromolecular crystals obey such decompositions. We tabulate these decompositions for those Sohncke groups that admit such decompositions. This has implications to the study of packing arrangements in macromolecular crystals. We also observe that every Sohncke group can be written as a product of Bieberbach and symmorphic subgroups, and this has implications for new nomenclature for space groups.

scholarly journals A remark on operating groups

1997 ◽  
Vol 20 (2) ◽  
pp. 393-395
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Short Note ◽  
Invariant Subgroup ◽  
Subnormal Subgroup ◽  
Operator Group ◽  
A Finite Group

LetGbe a finite group andHbe an operator group ofG. In this short note, we show a relationship between subnormal subgroup chains andH-invariant subgroup chains. We remark that the structure ofHis quite restricted whenGhas a specialH-invariant subgroup chain.

scholarly journals On supplements in finite groups

1963 ◽  
Vol 3 (1) ◽  
pp. 63-67
Author(s):  
R. Kochendörffer
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Factor Group ◽  
Extension Theory ◽  
A Finite Group

Let G be a finite group. If N denotes a normal subgroup of G, a subgroup S of G is called a supplement of N if we have G = SN. For every normal subgroup of G there is always the trivial supplement S = G. The existence of a non-trivial supplement is important for the extension theory, i.e., for the description of G by means of N and the factor group G/N. Generally, a supplement S is the more useful the smaller the intersection S ∩ N. If we have even S ∩ N = 1, then S is called a complement for N in G. In this case G is a splitting extension of N by S.

scholarly journals Proficient presentations and direct products of finite groups

1999 ◽  
Vol 60 (2) ◽  
pp. 177-189 ◽  
Author(s):  
K.W. Gruenberg ◽  
L.G. Kovács
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Module Structure ◽  
Direct Products ◽  
The Difference ◽  
A Finite Group ◽  
Open Question

Let G be a finite group, F a free group of finite rank, R the kernel of a homomorphism φ of F onto G, and let [R, F], [R, R] denote mutual commutator subgroups. Conjugation in F yields a G-module structure on R/[R, R] let dg(R/[R, R]) be the number of elements required to generate this module. Define d(R/[R, F]) similarly. By an earlier result of the first author, for a fixed G, the difference dG(R/[R, R]) − d(R/[R, F]) is independent of the choice of F and φ; here it is called the proficiency gap of G. If this gap is 0, then G is said to be proficient. It has been more usual to consider dF(R), the number of elements required to generate R as normal subgroup of F: the group G has been called efficient if F and φ can be chosen so that dF(R) = dG(R/[R, F]). An efficient group is necessarily proficient; but (though usually expressed in different terms) the converse has been an open question for some time.

scholarly journals Automorphisms of Finite Groups and their Fixed-Point Groups

1969 ◽  
Vol 9 (3-4) ◽  
pp. 467-477 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Prime Order ◽  
Soluble Group ◽  
Point Groups ◽  
Derived Series ◽  
A Finite Group

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

scholarly journals Involutory Automorphisms of Groups of odd Order and Their Fixed Point Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 113-120 ◽  
Author(s):  
L. G. Kovács ◽  
G. E. Wall
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Odd Order ◽  
Point Groups ◽  
A Finite Group ◽  
Automorphisms Of Groups ◽  
Elementary Result

Let G be a finite group of odd order with an automorphism θ of order 2. (We use without further reference the fact, established by W. Feit and J. G. Thompson, that all groups of odd order are soluble.) Let Gθ denote the subgroup of G formed by the elements fixed under θ. It is an elementary result that if Gθ = 1 then G is abelian. But if we merely postulate that Gθ be cyclic, the structure of G may be considerably more complicated—indeed G may have arbitrarily large soluble length.

Finite groups having unique proper characteristic subgroups. I

1955 ◽  
Vol 51 (1) ◽  
pp. 25-36 ◽  
Author(s):  
D. R. Taunt
Keyword(s):  
Finite Group ◽  
Direct Product ◽  
Finite Groups ◽  
Simple Groups ◽  
Characteristic Subgroup ◽  
Direct Products ◽  
A Finite Group

It is well known that a characteristically-simple finite group, that is, a group having no characteristic subgroup other than itself and the identity subgroup, must be either simple or the direct product of a number of isomorphic simple groups. It was suggested to the author by Prof. Hall that finite groups possessing exactly one proper characteristic subgroup would repay attention. We shall call a finite group having a unique proper characteristic subgroup a ‘UCS group’. In the present paper we first give some results on direct products of isomorphic UCS groups, and then we consider in more detail one of the types of UCS groups which can exist, that consisting of groups whose orders are divisible by exactly two distinct primes.

STRICT INEQUALITIES FOR MINIMAL DEGREES OF DIRECT PRODUCTS

2009 ◽  
Vol 79 (1) ◽  
pp. 23-30 ◽  
Author(s):  
NEIL SAUNDERS
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Negative Integer ◽  
The Other ◽  
Direct Products ◽  
Counter Example ◽  
Strict Inequalities ◽  
A Finite Group

AbstractThe minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.

Sign in / Sign up

Export Citation Format

Share Document