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Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms.
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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.
Let T be a spanning tree for Y and define the fundamental group Γ to be the group generated by the vertex groups G x and elements y for each edge of Y with the following relations: y = y −1 if y is the edge y with the reverse orientation. y φ y,0 (x) y −1 = φ y,1 (x) for all x in G y. y = 1 if y is an edge in T. This definition is ...
The group G is a 2-group, that is, every element in G has finite order that is a power of 2. [1] The group G is periodic (as a 2-group) and not locally finite (as it is finitely generated). As such, it is a counterexample to the Burnside problem. The group G has intermediate growth. [2] The group G is amenable but not elementary amenable. [2]
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.
For example, if x, y and z are elements of a group G, then xy, z −1 xzz and y −1 zxx −1 yz −1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G, [1] or even in every group. [2] Words play an important role in the theory of free groups and presentations, and are central objects of study in ...