Infinite Groups. Linear Groups
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Chum, G. Hulpke and G. Rosenberger — Groups,Complexity, Cryptology, 5, , Engel and G. Coleman, B. Matrix groups consist of matrices together with matrix multiplication. The general linear group GL n , R consists of all invertible n -by- n matrices with real entries. The dihedral group example mentioned above can be viewed as a very small matrix group. Another important matrix group is the special orthogonal group SO n.
It describes all possible rotations in n dimensions. Via Euler angles , rotation matrices are used in computer graphics. Representation theory is both an application of the group concept and important for a deeper understanding of groups. A broad class of group representations are linear representations, i. A representation of G on an n - dimensional real vector space is simply a group homomorphism. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.
Given a group action, this gives further means to study the object being acted on. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups , especially locally compact groups. Galois groups were developed to help solve polynomial equations by capturing their symmetry features. Similar formulae are known for cubic and quartic equations , but do not exist in general for degree 5 and higher. The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial.
Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory —a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
A group is called finite if it has a finite number of elements. The number of elements is called the order of the group. For example, the symmetric group on 3 letters S 3 is the group consisting of all possible orderings of the three letters ABC , i. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group S N for a suitable integer N , according to Cayley's theorem. Parallel to the group of symmetries of the square above, S 3 can also be interpreted as the group of symmetries of an equilateral triangle.
In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group G the order of any finite subgroup H divides the order of G.
The Sylow theorems give a partial converse. The dihedral group discussed above is a finite group of order 8. The order of r 1 is 4, as is the order of the subgroup R it generates see above. The order of the reflection elements f v etc. Both orders divide 8, as predicted by Lagrange's theorem. Mathematicians often strive for a complete classification or list of a mathematical notion.
In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p , a prime number, are necessarily cyclic abelian groups Z p. Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory , they are group objects in a category , meaning that they are objects that is, examples of another mathematical structure which come with transformations called morphisms that mimic the group axioms. For example, every group as defined above is also a set, so a group is a group object in the category of sets.
Some topological spaces may be endowed with a group law. Such groups are called topological groups, and they are the group objects in the category of topological spaces. All of these groups are locally compact , so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case of real numbers for example:. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups , which are basic to number theory.
Lie groups in honor of Sophus Lie are groups which also have a manifold structure, i. A standard example is the general linear group introduced above: it is an open subset of the space of all n -by- n matrices, because it is given by the inequality. Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities.
They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation —as a model of space time in special relativity.
By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. In abstract algebra , more general structures are defined by relaxing some of the axioms defining a group. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks.
Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n -ary one i. With the proper generalization of the group axioms this gives rise to an n -ary group. See classification of finite simple groups for further information. Some authors therefore omit this axiom.
However, group constructions often start with an operation defined on a superset, so a closure step is common in proofs that a system is a group. The notions of torsion of a module and simple algebras are other instances of this principle. See prime element. Up to isomorphism, there are about 49 billion. They are interchanged when passing to the dual category. From Wikipedia, the free encyclopedia. Algebraic structure with one binary operation. This article is about basic notions of groups in mathematics. For a more advanced treatment, see Group theory.
Basic notions. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics. Finite groups. Discrete groups Lattices. Topological and Lie groups. Algebraic groups. Linear algebraic group Reductive group Abelian variety Elliptic curve.
Group -like. Ring -like. Lattice -like. Module -like. Module Group with operators Vector space. Algebra -like.
Main article: History of group theory. The axioms for a group are short and natural Yet somehow hidden behind these axioms is the monster simple group , a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists. Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
The associativity constraint deals with composing more than two symmetries: Starting with three elements a , b and c of D 4 , there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose a and b into a single symmetry, then to compose that symmetry with c.
The other way is to first compose b and c , then to compose the resulting symmetry with a. While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. An inverse element undoes the transformation of some other element.
Main article: Group homomorphism. Main article: Subgroup. Main article: Coset.