The character group of a topological group that is an abelian group consists of its continuous group homomorphisms to the circle group, with pointwise multiplication. For a finite group, all characters are continuous and their values are roots of unity, so this agrees with the character group of a finite abelian group.
The character group consists of the group homomorphisms . Pointwise multiplication makes it a finite abelian group isomorphic to .
A multiplicative character is a group homomorphism from a finite field's multiplicative group of a finite field to the complex unit circle. A generator of the cyclic multiplicative group of a finite field produces a character of every order dividing . Extend every such character to zero by value zero, including the trivial multiplicative character, when using cyclotomic indicator formulas.
When are nontrivial, changing variables in a product of Gauss sums of finite-field characters gives , hence . If at least one character is nontrivial and their product is trivial, or the other character is trivial, the absolute value is one. Thus all non-main terms in a two-character cyclotomic intersection count are .
Here is a nontrivial additive character of the finite field. For nontrivial , substituting in the squared absolute value givesThe inner sum is at and otherwise; multiplicative character orthogonality completes the identity.
Let be finite abelian and let be a character of a finite abelian group. To adjoin , let be the least positive integer with and choose a complex number of modulus one with . Then is well-defined and extends to . There are exactly choices, corresponding to the roots. Iterating proves character extension to , proves that has exactly characters, and proves separation of points by extending a primitive character of the cyclic subgroup generated by a nonzero point. No structure theorem is needed for this argument.
If is a subgroup of a finite abelian group, , and is the least positive integer with , then each character of a finite abelian group on has exactly extensions to . The displayed root choice makes the extension well-defined. Starting with the trivial subgroup and adjoining generators proves that the number of characters equals the group order without using a decomposition into cyclic groups.
A character of a finite abelian group is a homomorphism into the complex unit circle. Equivalently it is a linear character; finite order forces each character value to be a root of unity. These are the elements of the character group of a finite abelian group and form the orthogonal Fourier basis of functions on the group.
For a linear character of a finite group ,Indeed, if , the bijection gives , forcing the sum to vanish. The trivial character gives directly. The same argument applies to the restriction of a linear character of a larger group to a subgroup.
For a subgroup of a finite abelian group , its annihilator isThese are exactly the linear characters that descend to the quotient group . The character group of a finite abelian group has the same size as the group, so . The annihilator is what abelian hidden-subgroup Fourier sampling measures.
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A character group typically refers to a collection or set of characters that share certain characteristics or properties, often used in various contexts including literature, psychology, gaming, and social dynamics. Here are a few interpretations of the term: 1. **Literature and Media**: In storytelling, a character group can refer to a cast of characters that interact within a narrative. This can include protagonists, antagonists, and supporting characters who may have different roles, motivations, and relationships.