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The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation T {\displaystyle \mathbb {T} } for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1- torus .
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.
The 2-adic integers, with selected corresponding characters on their Pontryagin dual group. In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and ...
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.
The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1. In the simple case n = 1, the group U(1) corresponds to the circle group, isomorphic to the set of all complex numbers that have absolute value 1, under multiplication ...
The universal covering group of the circle group T is the additive group of real numbers (R, +) with the covering homomorphism given by the mapping R → T : x ↦ exp(2πix). The kernel of this mapping is isomorphic to Z. For any integer n we have a covering group of the circle by itself T → T that sends z to z n.
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H.This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
A topological group G, or a partial piece of a group like F above, is said to have no small subgroups if there is a neighbourhood N of e containing no subgroup bigger than {e}. For example, the circle group satisfies the condition, while the p -adic integers Z p as additive group does not, because N will contain the subgroups: p k Z p , for all ...