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n(n + 1)/2

which gives 3 in case n = 2, and 6 for n = 3.

1 Subgroup structure

The Euclidean group has as subgroups the group T of translations, and the orthogonal group O(n). Any element of E(n) is a product of a translation followed by an orthogonal transformation, in a unique way. From the point of view of group theory, one notices that T is a normal subgroup of E(n): for any translation t and any isometry u, we have

u⁻¹tu

again a translation (one can say, through a displacement that is u acting on the displacement of t).

Together, these facts imply that E(n) is the semidirect product of O(n) extended by T. In other words O(n) is (in the natural way) also the quotient group of E(n) by T.

Now SO(n), the special orthogonal group, is a subgroup of O(n), of indexSee Also Index, Washington, United States. An index (pl. indices, sometimes indexes) is a pointer (literally note that the index finger is the one which points, or indicates that takes you to information. Traditionally indices are found at the back of boo two. Therefore E(n) has a subgroup E⁺(n), also of index two, consisting of direct isometries. That is, isometries not involving a change of orientation; equally, those represented as a translation followed by a rotationThis article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation). Rotation is the movement of a body in such a way that any given point of that body remains at a constant distance from some other fixed poi, rather than a translation followed by some kind of reflectionThe term reflection (also spelt reflexion can refer to several different concepts: In mathematics, reflection is the transformation of a space. In physics, reflection is a wave phenomenon. In computer science, reflection is a programming language feature (in dimensions 2 and 3, these are the familiar reflections in a mirrorThis article is about the reflective surfaces. A mirror is a reflective surface that is smooth enough to be able to form an image. The best known example is the plane mirror that most people have at home. In it, a parallel beam of light changes its direct line or plane, which may be taken to include the originThe origin of something (from the Latin origo "beginning") is where it came from, in the sense of a physical location or a metaphysical source. In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. The most).

2 Relation to the affine group

The Euclidean group E(n) is a subgroup of the affine groupIn mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is a Lie group if K is the real or complex field. There is more than one convenient way to describe th for n dimensions, and in such a way as to respect the semidirect product structure of both groups. As a consequence, Euclidean group elements can also be represented as square matrices of size n + 1, as explained for the affine group.

In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometryIn geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. It is the geometry of affine space of a given dimension n over a field K''. The case where K is the real numbers gives an adequate idea of the c. All affine theorems apply; the extra factor is the notion of distance, from which angle can be deduced.

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Topics: Euclidean Group E''(''n Groups Affine Subgroup Translation...

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Euclidean geometryEuclidean geometry In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Euclidean geometry someti Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers Euclidean domainIn abstract algebra, a Euclidean domain (also called a Euclidean ring is a type of ring in which the Euclidean algorithm can be used. More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements Euclidean algorithmThe Euclidean algorithm (also called Euclid's algorithm is an algorithm to determine the greatest common divisor (gcd) of two integers. It is one of the oldest algorithms known, since it appeared in Euclid's Elements around 300 BC. The algorithm does not Euclidean distanceThe Euclidean distance of two points x x . x and y y . y in Euclidean n space is computed as : It is the "ordinary" distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem Euclidean groupIn mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. It is therefore one of the oldest and most studied groups, at least in the cases of dimension 2 and 3 — implicitly, many of its properties are familiar, if not i Euclidean quantum gravity Euclidean minimum spanning tree