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Group theory - Biomatics.org

Group theory

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In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and operation must satisfy a few conditions called group axioms, namely associativity, identity and invertibility. While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a group—the formulation of the axioms is detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.

Contents

Examples of groups

 

 

The Integers form a group under the operation of addition. 0 is the identity and the inverse of an element is called its negative.

The set of Non-zero Rational Numbers with the group operation multiplication. In this group 1 is the identity and  the inverse is called the reciprocal.

The Positive Rational Numbers also form a group under multiplication.

 The set of Negative Rational Numbers does not form a group under multiplication since it not only is not closed but also does not contain an identity,  nor inverses

 

 



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QuCayleyColor.jpg 

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Symmetry groups

Coxeter group

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In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections.

Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.



 

Wallpaper Groups

A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinct groups.

 

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups).



Wallpaper Groups

 


 

Quad Tree

A quadtree is a tree data structure in which each internal node has up to four children. Quadtrees are most often used to partition a two dimensional space by recursively subdividing it into four quadrants or regions. The regions may be square or rectangular, or may have arbitrary shapes. This data structure was named a quadtree by Raphael Finkel and J.L. Bentley in 1974. A similar partitioning is also known as a Q-tree. All forms of Quadtrees share some common features:

 

  • They decompose space into adaptable cells
  • Each cell (or bucket) has a maximum capacity. When maximum capacity is reached, the bucket splits
  • The tree directory follows the spatial decomposition of the Quadtree

A point quadtree
 

A point quadtree



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Group Action

In mathematics, a symmetry group describes all symmetries of objects. This is formalized by the notion of a group action: every element of the group "acts" like a bijective map (or "symmetry") on some set. In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices, and is usually considered in the finite-dimensional case—it is the same as a group action of G on an ordered basis of a vector space.



Group Orbit

In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group G acts on a set X (this process is called a group action), it permutes the elements of X. Any particular element X moves around in a fixed path which is called its orbit. In the notation of set theory, the group orbit of a group element x can be defined as

 G(x)={gx in X:g in G},
(1)

where g runs over all elements of the group G. For example, for the permutation group G_1={(1234),(2134),(1243),(2143)}, the orbits of 1 and 2 are {1,2} and the orbits of 3 and 4 are {3,4}.

A group fixed point is an orbit consisting of a single element, i.e., an element that is sent to itself under all elements of the group. The stabilizer of an element x consists of all the permutations of G that produce group fixed points in x, i.e., that send x to itself. The stabilizers of 1 and 2 under G_1 are therefore {(1234),(1243)}, and the stabilizers of 3 and 4 are {(1234),(2134)}.

Note that if y in G(x) then x in G(y), because y=gx iff x=g^(-1)y. Consequently, the orbits partition X and, given a permutation group G on a set S, the orbit of an element s in S is the subset of S consisting of elements to which some element G can send s.

Space Groups

The space group of a crystal or crystallographic group is a mathematical description of the symmetry inherent in the structure. The word 'group' in the name comes from the mathematical notion of a group, which is used to build the set of space groups.

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, and the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion). Furthermore one must consider the screw axis and glide plane symmetry operations. These are called compound symmetry operations and are combinations of a rotation or reflection with a translation less than the unit cell size. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries.

 

Hexagonal Space Groups


In crystallography, the hexagonal crystal system is one of the 7 crystal systems, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families. They are closely related and often confused with each other, but they are not the same. The hexagonal lattice system consists of just one Bravais lattice type: the hexagonal one. The hexagonal crystal system consists of the 7 point groups such that all their space groups have the hexagonal lattice as underlying lattice. The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.[1]

 

 

Transformation Groups

Consider "addition" by examining Rubik's Cube. If you consider each valid transformation (which consists of a sequence of rotations of cross sections) an object, then we have a natural way of "adding" two objects, i.e. given two transformations, we first perform one, and then follow that by performing the other. One can check that this "addition" satisfies properties 0-3. For example, there is an identity transformation where you do nothing to the cube. Also, for any valid transformation, which consists of a sequence of rotations, we can undo the rotations one-by-one to arrive back at the identity transformation. However, this "addition" does not satisfy property 4, the commutative law. Given two transformation S and T, we may not have S+T = T+S. For example, if we rotate one vertical cross section up, then rotate a horizontal cross section right, we would get something different than the result of performing the two operations in the reverse order.

 


Groups, Rings, and Fields

Everyone is familiar with the basic operations of arithmetic, addition, subtraction, multiplication, and division. In the "new math" introduced during the 1960s in the junior high grades of 7 through 9, students were exposed to some mathematical ideas which formerly were not part of the regular school curriculum. Elementary set theory was one of them. Another was abstract algebra, in which it was illustrated that operations resembling addition and multiplication could exist that had some of their properties.

 

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