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Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law . A group and its underlying set are thus two different mathematical objects.
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 ...
A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie , who laid the foundations of the theory of continuous transformation groups .
Sociology. In the social sciences, a social group is defined as two or more people who interact with one another, share similar characteristics, and collectively have a sense of unity. [1] [2] Regardless, social groups come in a myriad of sizes and varieties. For example, a society can be viewed as a large social group.
Topological definition. A Lie group can be defined as a ( Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. [14] First, we define an immersely linear Lie group to be a subgroup G of the general linear group such that.
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory . Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general ...
For prime n the group is cyclic, and in general the structure is easy to describe, but no simple general formula for finding generators is known. Group axioms [ edit ] It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group .
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G . For a finite cyclic group G of order n we have G = {e, g, g2, ... , gn−1}, where e is the identity element and gi = gj whenever i ≡ j ( mod n ); in particular gn = g0 = e, and g−1 = gn−1.