Editing Character table (section)

==Properties==
Complex conjugation acts on the character table: since the complex conjugate of a representation is again a representation, the same is true for characters, and thus a character that takes on non-[[real number|real]] complex values has a conjugate character.

Certain properties of the group ''G'' can be deduced from its character table:

* The order of ''G'' is given by the sum of the squares of the entries of the first column (the degrees of the irreducible characters). More generally, the sum of the squares of the [[Complex number#Polar complex plane|absolute values]] of the entries in any column gives the order of the centralizer of an element of the corresponding conjugacy class.
*All [[normal subgroup]]s of ''G'' (and thus whether or not ''G'' is [[simple group|simple]]) can be recognised from its character table. The [[kernel (algebra)|kernel]] of a character χ is the set of elements ''g'' in ''G'' for which χ(g) = χ(1); this is a normal subgroup of ''G''. Each normal subgroup of ''G'' is the [[intersection (set theory)|intersection]] of the kernels of some of the irreducible characters of ''G''. 
*The number of irreducible representations of ''G'' equals the number of conjugacy classes that ''G'' has.
*The [[commutator subgroup]] of {{mvar|G}} is the intersection of the kernels of the linear characters of {{mvar|G}}.
*If {{mvar|G}} is finite, then since the character table is square and has as many rows as conjugacy classes, it follows that {{mvar|G}} is [[abelian group|abelian]] iff each conjugacy class has size 1 iff the character table of {{mvar|G}} is <math>|G| \!\times\! |G|</math> iff each irreducible character is linear.
*It follows, using some results of [[Richard Brauer]] from [[modular representation theory]], that the [[prime divisor]]s of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of [[Graham Higman]]).

The character table does not in general determine the group [[up to]] [[group isomorphism|isomorphism]]: for example, the [[quaternion group]] and the [[dihedral group]] of order 8 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 finite group up to isomorphism. In 1964, this was answered in the negative by [[E. C. Dade]].

The linear representations of {{mvar|G}} are themselves a group under the [[tensor product]], since the tensor product of {{nowrap|1-dimensional}} vector spaces is again {{nowrap|1-dimensional}}. That is, if <math>\rho_1:G \to V_1</math> and <math>\rho_2:G \to V_2</math> are linear representations, then <math>\rho_1\otimes\rho_2(g) = (\rho_1(g)\otimes\rho_2(g))</math> defines a new linear representation. This gives rise to a group of linear characters, called the [[character group]] under the operation <math>[\chi_1*\chi_2](g) = \chi_1(g)\chi_2(g)</math>. This group is connected to [[Dirichlet character]]s and [[Fourier analysis]].