Amenable groupIn mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".
Abelian groupIn mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
Finitely generated abelian groupIn abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate . Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers, , are a finitely generated abelian group. The integers modulo , , are a finite (hence finitely generated) abelian group.
Abelian extensionIn abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension.
Field extensionIn mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
Torsion-free abelian groupIn mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order. While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case. Abelian group An abelian group is said to be torsion-free if no element other than the identity is of finite order.
Category of abelian groupsIn mathematics, the Ab has the abelian groups as and group homomorphisms as morphisms. This is the prototype of an : indeed, every can be embedded in Ab. The zero object of Ab is the trivial group {0} which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a of Grp, the .
Free abelian groupIn mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis.
Abelian varietyIn mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field.
Purely inseparable extensionIn algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions. An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial.
Divisible groupIn mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups. An abelian group is divisible if, for every positive integer and every , there exists such that .
Separable extensionIn field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
Group extensionIn mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence If is an extension of by , then is a group, is a normal subgroup of and the quotient group is isomorphic to the group . Group extensions arise in the context of the extension problem, where the groups and are known and the properties of are to be determined.
Geometric group theoryGeometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects.
Representation theory of finite groupsThe representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero.
QuestionA question is an utterance which serves as a request for information. Questions are sometimes distinguished from interrogatives, which are the grammatical forms typically used to express them. Rhetorical questions, for instance, are interrogative in form but may not be considered bona fide questions, as they are not expected to be answered. Questions come in a number of varieties. Polar questions are those such as the English example "Is this a polar question?", which can be answered with "yes" or "no".
Burnside ringIn mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967). Given a finite group G, the generators of its Burnside ring Ω(G) are the formal sums of isomorphism classes of finite G-sets. For the ring structure, addition is given by disjoint union of G-sets and multiplication by their Cartesian product.
Alexander GrothendieckAlexander Grothendieck (ˈgroʊtəndiːk; ˌalɛˈksandɐ ˈɡʁoːtn̩ˌdiːk; ɡʁɔtɛndik; 28 March 1928 – 13 November 2014) was a French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century.
Nicolas BourbakiNicolas Bourbaki (nikɔla buʁbaki) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the Éléments de mathématique (Elements of Mathematics), the group's central work.
