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Arthur Cayley's On the theory of groups, as depending on the symbolic equation = (1854) gives the first abstract definition of a finite group. [14] Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. [15] After novel geometries such as hyperbolic and ...
Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of ...
Geometric 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 can act non-trivially (that is, when the groups in question are realized as geometric symmetries or ...
An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes. A localic group is a group object in the category of locales. The group objects in the category of groups (or monoids) are the abelian groups.
In mathematics, a Euclidean group is the group of ... The origin of Euclidean geometry allows definition of the notion of distance, from which angle can then be deduced.
The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry.
The group consists of the finite strings (words) that can be composed by elements from A, together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance (abb) • (bca) = abbbca. Every group (G, •) is basically a factor group of a free group generated by G.
A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
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