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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.
Characters are invariant on conjugacy classes. The characters of irreducible representations are orthogonal. The primary importance of the character group for finite abelian groups is in number theory, where it is used to construct Dirichlet characters. The character group of the cyclic group also appears in the theory of the discrete Fourier ...
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 property of X). Similarly, a set of properties P is said to characterize X, when these properties distinguish X from all other objects. Even though a characterization identifies ...
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 ...
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 ...
In mathematics, for a function :, the image of an input value is the single output value produced by when passed .The preimage of an output value is the set of input values that produce .
The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each
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.