Finite group - Citizendia

In mathematics, a finite group is a group which has finitely many elements. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. The twentieth century of the Common Era began on In Mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each Prime number p first In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the It is too much to hope for a complete determination of the structure of all finite groups: the number of possible structures soon becomes overwhelming. However, the twentieth century saw the classification of the finite simple groups, which may be viewed as the determination of the "building blocks" for all finite groups, as each finite group has a composition series. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces

Thanks to the work of mathematicians such Chevalley and Steinberg, the second half of the twentieth century also saw increased understanding of finite analogs of classical groups, and other related groups. Claude Chevalley ( 11 February 1909, Johannesburg, South Africa - 28 June 1984, Paris) was a French Robert Steinberg (born May 25[[ 922]] Soroki, Romania) is a mathematician at the University of California Los Angeles who invented the The classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces There is a certain leeway in using the term One such family of groups is the family of general linear groups over finite fields. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements The group theorist J. L. Alperin has written that "The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled. "^[1]

Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure preserving transformations. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetries as motions as opposed to e In Mathematics, in particular the theory of Lie algebras the Weyl group of a Root system &Phi is a Subgroup of the Isometry group These are finite groups generated by reflections which act on a finite dimensional Euclidean space. Thus properties of finite groups can play a role in subjects such as theoretical physics. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world

Number of groups of a given order

Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of If n is the square of a prime, then there are exactly two possible isomorphism types of group of order n, both of which are abelian. If n is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order n, and the number grows very rapidly as the power increases. Graham Higman FRS ( 19 January 1917 &ndash 8 April 2008) was a leading British Mathematician.

Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which For example, every group of order pq is cyclic when p and q are different primes with q less than p and p-1 not divisible by q. If n is squarefree, then any group of order n is solvable. A theorem of William Burnside, proved using group characters, states that every group of order n is solvable when n is divisible by fewer than three distinct primes. William Burnside ( July 2 1852 - August 21 1927) was an English Mathematician. This article refers to the use of the term character theory in mathematics for the media studies definition see Character theory (Media. By the Feit-Thompson theorem, which has a long and complicated proof, every group of order n is solvable when n is odd. In Mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable.

There is a meaningful sense in which for every positive integer n, most groups of order n are solvable. To see this for any particular order is usually not difficult (for example, there is (up to isomorphism) only one non-solvable group of order 60, while there are two non-isomorphic abelian groups of order 60 and several more isomorphism types of non-abelian solvable groups of order 60) but to make such a statement precise for all n requires the classification of finite simple groups. The classification of the finite simple groups, also called the enormous theorem is believed to classify all finite simple groups. Without the classification theorem, it is not clear whether there is a constant bounding the number of isomorphism types of simple groups of order n (with the benefit of the classification, it is known that the constant 2 is an upper bound for all n. Prior to the classification, it had long been known that there were infinitely many values of n for which two non-isomorphic simple groups of order n existed).

References

^ Jonathan L. Alperin, Book review: B. Huppert and N. Blackburn Title: Finite groups, Bulletin (New Series) of the American Mathematical Society, 10 (1984) 121, doi:10.1090/S0273-0979-1984-15210-8

Number of groups of a given order

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