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In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem. Kernel of a group homomorphism. It is the preimage of the identity in the codomain of a group homomorphism. Every normal subgroup is the kernel of a group ...
Group dynamics is a system of behaviors and psychological processes occurring within a social group (intragroup dynamics), or between social groups (intergroup dynamics). The study of group dynamics can be useful in understanding decision-making behaviour, tracking the spread of diseases in society, creating effective therapy techniques, and ...
Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that = {}).Then is called a lattice in if in addition there exists a Borel measure on the quotient space / which is finite (i.e. (/) < +) and -invariant (meaning that for any and any open subset / the equality () = is satisfied).
In detail, the space of homomorphisms from G to the (cyclic) group of order p, (, /), is a vector space over the finite field = /. A non-trivial such map has as kernel a normal subgroup of index p, and multiplying the map by an element of ( Z / p ) × {\displaystyle (\mathbf {Z} /p)^{\times }} (a non-zero number mod p ) does not change the ...
The braid group B 3 is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group SL 2 (R) → PSL 2 (R). Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of B 3 modulo its center ; equivalently, to the ...
In this case, the models should be independent of the specific details of the task that the group is performing. On the other hand, some models might describe phases of the group's task performance and, because of this, tend to be very sensitive to the type of task that the group is engaged in (the "acting system", [3] p. 101).