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...

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

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

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

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

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

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...

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

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]....

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

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...

alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree...

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...

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...

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical...

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...

In abstract algebra, a cyclic group or monogenous group is a group, denoted C n {\displaystyle C_{n}} (also frequently Z n {\displaystyle \mathbb {Z} _{n}}...

(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...

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that...

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

mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at...

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

In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations...

mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points...

finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called...

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

finite groups, or just the sporadic groups. A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself...

Not all algebraic groups are linear groups or abelian varieties; for instance, some group schemes occurring naturally in arithmetic geometry are neither...