In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example S L 2 ( Z ) . {\displaystyle \mathrm {SL}...

Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic...

congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important...

mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap...

Margulis arithmeticity theorem says, in particular: for a simple Lie group G of real rank at least 2, every lattice in G is an arithmetic group. In seeking...

mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders...

lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac–Moody...

Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider...

work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by...

The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It...

In abstract algebra, a cyclic group or monogenous group is a group, denoted Cn (also frequently Z {\displaystyle \mathbb {Z} } n or Zn, not to be confused...

specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently...

position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities. Modular arithmetic for a modulus n {\displaystyle...

the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the...

(this results from the fundamental theorem of arithmetic). The center Z ( G ) {\displaystyle Z(G)} of a group G {\displaystyle G} is the set of elements...

In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces...

Bibcode:2023arXiv230414646D. doi:10.1016/j.aim.2025.110214. Duncan, John F. (2008). "Arithmetic groups and the affine E8 Dynkin diagram". arXiv:0810.1465 [RT math. RT]....

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

Quaternionic, Fermionic". Retrieved 1 February 2012. Milne, Algebraic Groups and Arithmetic Groups, p. 103 Bak, Anthony (1969), "On modules with quadratic forms"...

performed using modular arithmetic with modulus n. If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations...

In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible...

finite. crystallographic point group congruence subgroup arithmetic group geometric group theory computational group theory freely discontinuous free...

mathematics, a group action of a group G {\displaystyle G} on a set S {\displaystyle S} is a group homomorphism from G {\displaystyle G} to some group (under...

In mathematics, the general linear group of degree n {\displaystyle n} is the set of n × n {\displaystyle n\times n} invertible matrices, together with...

In group theory, the Tits group 2F4(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order    17,971,200 = 211 · 33 · 52 · 13...

mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for...

{suv}}=2\,\Re {\mathord {\bigl (}}\,suv\,{\bigr )}\;,} complex number arithmetic shows | u z + v | 2 = S + z z ∗  and  | v ∗ z + u ∗ | 2 = S + 1   , {\displaystyle...

the relation to moduli spaces, and not from modular arithmetic. The modular group Γ is the group of fractional linear transformations of the complex upper...

In geometry and group theory, a lattice in the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} is an infinite set of points in this space with...

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...