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The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
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 finite abelian group of order n has exactly n distinct characters. These are denoted by f 1, ..., f n. The function f 1 is the trivial representation, which is given by () = for all . It is called the principal character of G; the others are called the non-principal characters.
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. This group is referred to as the ...
Furthermore, a class function on is a character of if and only if it can be written as a linear combination of the distinct irreducible characters with non-negative integer coefficients: if is a class function on such that = + + where non-negative integers, then is the character of the direct sum of the representations corresponding to .
Data reduction is the transformation of numerical or alphabetical digital information derived empirically or experimentally into a corrected, ordered, and simplified form. . The purpose of data reduction can be two-fold: reduce the number of data records by eliminating invalid data or produce summary data and statistics at different aggregation levels for various applications
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A fundamental example of a reductive group is the general linear group of invertible n × n matrices over a field k, for a natural number n. In particular, the multiplicative group G m is the group GL (1), and so its group G m ( k ) of k -rational points is the group k * of nonzero elements of k under multiplication.