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For a subgroup $K$ of a finite abelian group $G$, its annihilator isThese are exactly the linear characters that descend to the quotient group $G/K$. The character group of a finite abelian group has the same size as the group, so $|K^{\perp }|=|G/K|=|G|/|K|$. The annihilator is what abelian hidden-subgroup Fourier sampling measures.

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 $\chi$ of a finite group $K$,Indeed, if $\chi (h)\ne 1$, the bijection $k\mapsto hk$ gives ${\sum }_k\chi (k)=\chi (h){\sum }_k\chi (k)$, forcing the sum to vanish. The trivial character gives $|K|$ directly. The same argument applies to the restriction of a linear character of a larger group to a subgroup.

If a finite group has no nontrivial homomorphism to ${\mathbb{C}}^{\times }$, its polynomial invariant ring is a unique factorization domain. Factor an invariant in the ambient polynomial ring and collect its irreducible factors into orbit products. Each orbit product transforms by a linear character, hence is invariant. It is prime in the invariant ring, and the invariant factorization is a product of these primes.

For a finite abelian group $G$, $U(h)|g⟩=|g+h⟩$ defines the regular representation on the basis labelled by $G$. The linear characters give the common eigenbasisCharacter orthogonality proves orthonormality, and changing variables by the translation in a group proves the eigenvalue formula.

Forthere are $2n$ linear characters, given by $a\mapsto {\xi }^j$, $b\mapsto 1$ for a primitive $(2n)$th root of unity $\xi$, and $n$ two-dimensional irreducible representationswhere ${\omega }^3=1$ and $\omega \ne 1$. The two-dimensional representations parametrized by $\epsilon$ and $-\epsilon$ are isomorphic. The sum of squares of irreducible degrees is $2n+4n=6n$, so this list is complete.