Group theory is the study of algebraic structures called groups. This introduction will rely heavily on set theory and modular arithmetic as well. Later on it will require an understanding of mathematical induction, functions, bijections, and partitions. Lessons may utilize matricies and complex numbers as well. What is a group? A more rigorous definition will come shortly but to give a very rough idea of a group it is a set and a operation.
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All of these structures have things in common; they are all integral to being groups. They also have things in common that aren't necessary to groups. Let's examine some of these similarities. All of these groups have a closed binary operation. This is called closure. Notice that all of the groups in the above examples are closed under their respective operations. This is called associativity and is required for a structure to be a group. Notice that for any integer m.
Zero is the only element in this group with this property and it's called the identity of the group. In all of the above examples the underlying set of the groups are infinite, but groups need not be infinite. Note that with the requirement of an identity element the underlying set cannot be the empty set. All of the groups above are commutative. This is not true of all groups in general. Groups that are commutative are called Abelian Groups. This set is closed but it doesn't have inverses therefore it is not a group.
Consider the set of all matricies under addition. This is not a group because not all matricies can be added. Consider for example a 2x2 matrix and a 3x3 matrix. Since groups are associative it is common place to drop the parentheses when one is working with something shown to be a group. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation.
The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines , one can show that there is in general no algorithm solving this task.
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Another, generally harder, algorithmically insoluble problem is the group isomorphism problem , which asks whether two groups given by different presentations are actually isomorphic. Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.
Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X , for example a compact manifold , then G is quasi-isometric i.
Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry.
The identity keeping the object fixed is always a symmetry of an object.
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Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object. The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms , and the symmetry group is the automorphism group of the object in question.
Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings , for example, can be viewed as abelian groups corresponding to addition together with a second operation corresponding to multiplication.
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Therefore, group theoretic arguments underlie large parts of the theory of those entities. Galois theory uses groups to describe the symmetries of the roots of a polynomial or more precisely the automorphisms of the algebras generated by these roots. The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory.
It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S 5 , the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory.
Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different.
The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg—MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups.
Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. Algebraic geometry likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. They are both theoretically and practically intriguing. Toroidal embeddings have recently led to advances in algebraic geometry , in particular resolution of singularities. Algebraic number theory makes uses of groups for some important applications.
For example, Euler's product formula ,. The failure of this statement for more general rings gives rise to class groups and regular primes , which feature in Kummer's treatment of Fermat's Last Theorem. Analysis on Lie groups and certain other groups is called harmonic analysis.
Haar measures , that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques. In combinatorics , the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma. The presence of the periodicity in the circle of fifths yields applications of elementary group theory in musical set theory.
In physics , groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem , every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories.
In chemistry and materials science , groups are used to classify crystal structures , regular polyhedra, and the symmetries of molecules. Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane.
In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out.
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The symmetry operations of a molecule determine the specific point group for this molecule. In chemistry , there are five important symmetry operations. The identity operation E consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. Rotation around an axis C n consists of rotating the molecule around a specific axis by a specific angle. Other symmetry operations are: reflection, inversion and improper rotation rotation followed by reflection.
Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs , relating to the summing of an infinite number of probabilities to yield a meaningful solution.
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Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate.
One of the earliest encryption protocols, Caesar's cipher , may also be interpreted as a very easy group operation. Most cryptographic schemes use groups in some way. In particular Diffie—Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.