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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.
The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements, D 4, have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a ...
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.
For example, if p is prime and q(X) is an irreducible polynomial with coefficients in the field with p elements, then the quotient ring [] / (()) is a field of characteristic p. Another example: The field C {\displaystyle \mathbb {C} } of complex numbers contains Z {\displaystyle \mathbb {Z} } , so the characteristic of C {\displaystyle \mathbb ...
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 .
In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
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