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An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes. A localic group is a group object in the category of locales. The group objects in the category of groups (or monoids) are the abelian groups.
Even if the shapes, sizes, and objects are radically different, they will appear as a group if they are close. Refers to the way smaller elements are "assembled" in a composition. Also called "grouping", the principle concerns the effect generated when the collective presence of the set of elements becomes more meaningful than their presence as ...
Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. [7] The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group.
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.
Categorization is a type of cognition involving conceptual differentiation between characteristics of conscious experience, such as objects, events, or ideas.It involves the abstraction and differentiation of aspects of experience by sorting and distinguishing between groupings, through classification or typification [1] [2] on the basis of traits, features, similarities or other criteria that ...
For example, in the figure illustrating the law of proximity, there are 72 circles, but we perceive the collection of circles in groups. Specifically, we perceive that there is a group of 36 circles on the left side of the image and three groups of 12 circles on the right side of the image.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation.
The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations.