Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has...

a discrete subgroup of PSL(2,R). The modular group is a lattice in PSL(2,R), but it is not cocompact. Kleinian groups are, by definition, discrete subgroups...

points Lattice (discrete subgroup), a discrete subgroup of a topological group whose quotient carries an invariant finite Borel measure Lattice (module)...

but the rational numbers, Q, do not. A lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space...

Congruence relation Equivalence class Equivalence relation Lattice (group) Lattice (discrete subgroup) Multiplication table Prime number Up to Abelian variety...

arithmetic lattice in S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbb {R} )} . A lattice in a Lie group is usually defined as a discrete subgroup with...

algorithms for factoring and finding discrete logarithms in quantum computing are instances of the hidden subgroup problem for finite abelian groups, while...

derivative Darboux derivative Lie groupoid Lie algebroid Lattice (group) Lattice (discrete subgroup) Frieze group Wallpaper group Space group Crystallographic...

some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the...

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...

have infinite order. The subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. (While the infimum here...

homomorphism is just the fiber over the identity in H and is a discrete normal subgroup of G. The kernel K is closed in G if and only if G is Hausdorff...

theorem states that every group G {\displaystyle G} is isomorphic to a subgroup of the symmetric group on (the underlying set of) G {\displaystyle G} ...

In mathematics, especially in geometry, a double lattice in ℝn is a discrete subgroup of the group of Euclidean motions that consists only of translations...

modular lattices are the lattice of submodules of a module (hence modular), the lattice of two-sided ideals of a ring, and the lattice of normal subgroups of...

reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite...

group WK of the maximal compact subgroup K. If we fix a fundamental chamber for the Weyl group of K, then the discrete series representation are in 1:1...

mathematics, the term lattice group is used for two distinct notions: a lattice (group), a discrete subgroup of Rn and its generalizations a lattice ordered group...

example, SL(n,Z) is an arithmetic subgroup of SL(n,Q). For a Lie group G, a lattice in G means a discrete subgroup Γ of G such that the manifold G/Γ has...

the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of automorphisms of the Leech lattice, and shows that it has two complex irreducible...

If 0 is removed from this structure it remains a lattice but ceases to be complete. The subgroups of any given group under inclusion. (While the infimum...

\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\displaystyle d\leq n} , the LLL algorithm...

constructing subgroups of semisimple Lie groups produces examples of lattices, called arithmetic lattices. It is analogous to considering the subgroup SL(n,Z)...

Bethe lattices also occur as the discrete subgroups of certain hyperbolic Lie groups, such as the Fuchsian groups. As such, they are also lattices in the...

space, and its closed subgroup L^ dual to L turns out to be a lattice in V^. Therefore, L^ is the natural candidate for dual lattice, in a different vector...

mod 3, so is a subgroup of the Chevalley group E8(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson...

equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation. Analogously, an operator A on functions...

solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof...

doi:10.1016/0021-8693(69)90113-6, MR0251133, ISSN 0021-8693 MathWorld: Janko Groups Atlas of Finite Group Representations: J2 The subgroup lattice of J2...

preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of E +(m) (see Euclidean group). Symmetry with respect to all rotations...