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Characteristic classes are phenomena of cohomology theory in an essential way — they are contravariant constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section one does need that variance. In fact cohomology theory grew up after homology and homotopy theory ...
In mathematics, a characterization of an object is a set of conditions that, while possibly different from the definition of the object, is logically equivalent to it. [1] To say that "Property P characterizes object X" is to say that not only does X have property P, but that X is the only thing that has property P (i.e., P is a defining ...
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch( G ) of these morphisms forms an abelian group under pointwise multiplication.
A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers , and the class of all sets, are proper classes in many formal systems.
Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in the classification of finite simple groups. Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values.
In mathematics, specifically category theory, a functor is a mapping between categories.Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces.
In mathematics, a character group is the group of representations of an abelian group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters that arise in the related context of character theory .
This is the appropriate partial ordering because of such facts as that char(A × B) is the least common multiple of char A and char B, and that no ring homomorphism f : A → B exists unless char B divides char A. The characteristic of a ring R is n precisely if the statement ka = 0 for all a ∈ R implies that k is a multiple of n.