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In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
It is straightforward to generalize the notion of a residue-class-wise affine group to groups acting on suitable rings other than , though only little work in this direction has been done so far. See also the Collatz conjecture , which is an assertion about a surjective , but not injective residue-class-wise affine mapping.
The regular triangular tiling of the plane, whose symmetries are described by the affine symmetric group S̃ 3. The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling of the plane, as well as related higher-dimensional objects.
Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue. The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An ...
The additive group: the affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its k {\displaystyle k} -points are isomorphic as a group to the additive group of k {\displaystyle k} ), and usually denoted by G a {\displaystyle \mathrm {G} _{a}} .
Canadian Open Mathematics Challenge — Canada's premier national mathematics competition open to any student with an interest in and grasp of high school math and organised by Canadian Mathematical Society; Canadian Mathematical Olympiad — competition whose top performers represent Canada at the International Mathematical Olympiad
In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of , so the set of fixed points of forms an affine space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).
For example, the group of projective geometry in n real-valued dimensions is the symmetry group of n-dimensional real projective space (the general linear group of degree n + 1, quotiented by scalar matrices). The affine group will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity.