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This sum can help narrow down the dimensions of the irreducible representations in a character table. For example, if the group has order 10 and 4 conjugacy classes (for instance, the dihedral group of order 10) then the only way to express the order of the group as a sum of four squares is = + + +, so we know the dimensions of all the ...
The space of complex-valued class functions of a finite group G has a natural inner product: , := | | () ¯ where () ¯ denotes the complex conjugate of the value of on g.With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table:
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.
The finite group notation used is: Z n: cyclic group of order n, D n: dihedral group isomorphic to the symmetry group of an n–sided regular polygon, S n: symmetric group on n letters, and A n: alternating group on n letters. The character tables then follow for all groups. The rows of the character tables correspond to the irreducible ...
As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory [citation needed], combinatorics and number theory. Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an ...
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The conclusion from applying character theory to the group G is that G has the following structure: there are primes p>q such that (p q –1)/(p–1) is coprime to p–1 and G has a subgroup given by the semidirect product PU where P is the additive group of a finite field of order p q and U its elements of norm 1.