|
HNN Extension
In mathematics, the HNN extension is an important construction of combinatorial group theory. Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into another group ''G' '', in such a way that two given isomorphic subgroups of ''G'' are conjugate (through a given isomorphism) in ''G' ''. Construction Let ''G'' be a group with presentation G = \langle S \mid R\rangle , and let \alpha\colon H \to K be an isomorphism between two subgroups of ''G''. Let ''t'' be a new symbol not in ''S'', and define :G*_ = \left \langle S,t \mid R, tht^=\alpha(h), \forall h\in H \right \rangle. The group G*_ is called the ''HNN extension of'' ''G'' ''relative to'' α. The original group G is called the ''base group'' for the construction, while the subgroups ''H'' and ''K'' are the ''associated subgroups''. The new generator ''t'' is called the ''stable letter''. Key properties Since the presentation fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have Group isomorphism, isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface (mathematics), surface), and some point in it, and all the loops both starting and ending at this point—path (topology), paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then alo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word Problem For Groups
A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no consensus among linguists on its definition and numerous attempts to find specific criteria of the concept remain controversial. Different standards have been proposed, depending on the theoretical background and descriptive context; these do not converge on a single definition. Some specific definitions of the term "word" are employed to convey its different meanings at different levels of description, for example based on phonological, grammatical or orthographic basis. Others suggest that the concept is simply a convention used in everyday situations. The concept of "word" is distinguished from that of a morpheme, which is the smallest unit of language that has a meaning, even if it cannot stand on its own. Words are made out of at least one morpheme. Morphemes can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentation Of A Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Presented Group
In mathematics, a presentation is one method of specifying a group (mathematics), group. A presentation of a group ''G'' comprises a generating set of a group, set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is group isomorphism, isomorphic to the quotient group, quotient of a free group on ''S'' by the Normal closure (group theory), normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of such elements. By definition, every finite group is finitely generated, since ''S'' can be taken to be ''G'' itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated. Examples * Every quotient of a finitely generated group ''G'' is finitely generated; the quotient group is generated by the images of the generators of ''G'' under the canonical projection. * A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z. ** A locally cyclic group is a group in which every finitely gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higman's Embedding Theorem
In group theory, Higman's embedding theorem states that every finitely generated recursively presented group ''R'' can be embedded as a subgroup of some finitely presented group ''G''. This is a result of Graham Higman from the 1960s. On the other hand, it is an easy theorem that every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (up to isomorphism) exactly the finitely generated subgroups of finitely presented groups. Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups. As a corollary, there is a universal finitely presented group that contains ''all'' finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism). Higman's embedding theorem also implies the Novikov-Boone theorem (origi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Stillwell
John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University. Biography He was born in Melbourne, Australia and lived there until he went to the Massachusetts Institute of Technology for his doctorate. He received his PhD from MIT in 1970, working under Hartley Rogers, Jr, who had himself worked under Alonzo Church. From 1970 until 2001, he taught at Monash University back in Australia and in 2002 began teaching in San Francisco. Honors In 2005, Stillwell was the recipient of the Mathematical Association of America's prestigious Chauvenet Prize for his article "The Story of the 120-cell, 120-Cell," Notices of the AMS, January 2001, pp. 17–24. In 2012, he became a fellow of the American Mathematical Society. Works Books Stillwell is the author of many textbooks and other books on mathematics including: *''Classical Topology and Combinatorial Group Theory'', 1980, *2012 pbk reprint of 1993 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Of Groups
In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabilizer subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre. Definition A graph of groups over a graph is an assignment to each vertex of of a group and to each edge of of a group as well as monomorphisms and mapping into the groups assigned to the vertices at its ends. Fundamental group Let be a spanning tree for and define the fundamental group to be the group generated by the vertex groups and elements for each edge of w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bass–Serre Theory
Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory. History Bass–Serre theory was developed by Jean-Pierre Serre in the 1970s and formalized in ''Trees'', Serre's 1977 monograph (developed in collaboration with Hyman Bass) on the subject. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. However, the theory quickly became a standard tool of geometric group theory and geometric topology, particularly the study of 3-manifolds. Subsequent work of Bass contributed su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
