Editing Group (mathematics) (section)

== Groups with additional structure ==
An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set <math>G</math> equipped with a binary operation <math>G \times G \rightarrow G</math> (the group operation), a [[unary operation]] <math>G \rightarrow G</math> (which provides the inverse) and a [[nullary operation]], which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids [[existential quantifier]]s and is used in computing with groups and for [[computer-aided proof]]s.

This way of defining groups lends itself to generalizations such as the notion of [[group object]] in a category. Briefly, this is an object with [[morphism]]s that mimic the group axioms.{{sfn|Awodey|2010|loc=§4.1}}

=== Topological groups ===
[[Image:Circle as Lie group2.svg|right|thumb|The [[unit circle]] in the [[complex plane]] under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every [[Neighbourhood (mathematics)|small piece]], such as the red arc in the figure, looks like a part of the [[real line]] (shown at the bottom).|alt=A part of a circle (highlighted) is projected onto a line.]]
{{Main|Topological group}}
Some [[topological space]]s may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, <math>g \cdot h</math> and <math>g^{-1}</math> must not vary wildly if <math>g</math> and <math>h</math> vary only a little. Such groups are called ''topological groups,'' and they are the group objects in the [[category of topological spaces]].{{sfn|Husain|1966}} The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other [[topological field]], such as the field of complex numbers or the field of [[p-adic number|{{math|''p''}}-adic numbers]]<!-- Use math template rather than LaTeX markup to keep the linked text colored correctly -->. These examples are [[locally compact topological group|locally compact]], so they have [[Haar measure]]s and can be studied via [[harmonic analysis]]. Other locally compact topological groups include the group of points of an algebraic group over a [[local field]] or [[adele ring]]; these are basic to number theory{{sfn|Neukirch|1999}} Galois groups of infinite algebraic field extensions are equipped with the [[Krull topology]], which plays a role in [[Fundamental theorem of Galois theory#Infinite case|infinite Galois theory]].{{sfn|Shatz|1972}} A generalization used in algebraic geometry is the [[étale fundamental group]].{{sfn|Milne|1980}}

=== Lie groups ===
{{Main|Lie group}}
A ''Lie group'' is a group that also has the structure of a [[differentiable manifold]]; informally, this means that it [[diffeomorphism|looks locally like]] a Euclidean space of some fixed dimension.{{sfn|Warner|1983}} Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be [[smooth map|smooth]].

A standard example is the general linear group introduced above: it is an [[open subset]] of the space of all <math>n</math>-by-<math>n</math> matrices, because it is given by the inequality
<math display=block>\det (A) \ne 0,</math>
where <math>A</math> denotes an <math>n</math>-by-<math>n</math> matrix.{{sfn|Borel|1991}}

Lie groups are of fundamental importance in modern physics: [[Noether's theorem]] links continuous symmetries to [[conserved quantities]].{{sfn|Goldstein|1980}} [[Rotation]], as well as translations in [[space]] and [[time]], are basic symmetries of the laws of [[mechanics]]. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.{{efn|See [[Schwarzschild metric]] for an example where symmetry greatly reduces the complexity analysis of physical systems.}} Another example is the group of [[Lorentz transformation]]s, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of [[Minkowski space]]. The latter serves—in the absence of significant [[gravitation]]—as a model of [[spacetime]] in [[special relativity]].{{sfn|Weinberg|1972}} The full symmetry group of Minkowski space, i.e., including translations, is known as the [[Poincaré group]]. By the above, it plays a pivotal role in special relativity and, by implication, for [[quantum field theories]].{{sfn|Naber|2003}} [[Local symmetry|Symmetries that vary with location]] are central to the modern description of physical interactions with the help of [[gauge theory]]. An important example of a gauge theory is the [[Standard Model]], which describes three of the four known [[fundamental force]]s and classifies all known [[elementary particle]]s.{{sfn|Zee|2010}}