# Groups

In this subdiscipline: 269 papers

## Discipline summary

Start with a set and an operation acting on pairs formed from that set.
0) Assume every possible pair yields something from that operation.
1) Assume you can chain that operation without taking care in which order that operation is performed.
2) Assume there exists an element which leaves any other element invariant under that operation.
3) Assume for each element there exists an inverse: Another unique element, such that the operation applied to them in any order will result in the element leaving others invariant.

you get a structure composed of the set and a 2-parameter function representing the operation(0). That function or operation has a neutral element(2) and is associative(1). Additionally the function can be inverted in each parameter: Applying the operation to that parameter's inverse object(3) and the given output. Of course, applied in the same order as was the order of the parameters.

That's a group! It's called commutative when the order of the operator doesn't change the result. Groups can be thought of as sets of matrices, along with matrix multiplication. Another interesting point of view is bijective functions on some set of numbers...

## Popular papers

1. An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded…
2. These notes are a self-contained introduction to the use of dynamical and probabilistic methods in the study of hyperbolic groups. Most of this material is standard; however some of the proofs given are new, and some results are proved in greater…
3. Investigators have modeled oceanic and atmospheric vortices in the laboratory in a number of different ways, employing background rotation, density effects, and geometrical confinement. In this article, we address barotropic vortices in a rotating…
4. Let $\Gamma$ be a finitely generated group, and let S be a fi- nite, non-necessarily symmetric, generating subset of $\Gamma$. Let h be the transition operator of the directed Cayley graph \g(F,S), acting on l^2(F). Staring with Kesten's seminal…
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6. A Course in the Theory of Groups is a comprehensive introduction to the theory of groups - finite and infinite, commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the different…
7. The Poisson boundary of a group G with a probability measure \mu is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an integral…
8. Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey…
9. The Plancherel formula for the universal covering group of $SL(2, R)$ derived earlier by Pukanszky on which Herb and Wolf build their Plancherel theorem for general semisimple groups is reconsidered. It is shown that a set of unitarily equivalent…
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