I have a few related questions. First I would appreciate it if someone could provide me with a reference for the following


"Complex unitary irreducibles of virtually abelian groups have bounded degrees."


I am only concerned with complex unitary representations in this question.


Do we have a quantitative version of the above statement? For example, is the following true?


"If $G$ is a virtually abelian group with a finite index abelian subgroup H, then the degrees of irreducibles of $G$ are bounded above by $[G:H]$"


If H is additionally normal, is it true that the degrees of irreducibles of G divide $[G:H]$?


If these are not true in general, would they be true if $G$ is finitely generated in addition to being virtually abelian?



Irreducibles of virtually abelian finitely generated groups
I can at least explain how this bounded degree property works roughly when $G$ is a countable virtually abelian group. Kaplansky shows in his paper Groups with representations of bounded degree, Canadian J. Math vol. 1 (1949), 105-112 that the group ring of a virtually abelian group satisfies a polynomial identity.


If $K$ is any field, then the image of $KG$ under an irreducible representation is a primitive $K$-algebra satisfying this same polynomial identity. In Kaplansky, Rings with a polynomial identity, Bull. Amer. Math. Soc. 54 (1948) 575–580 that a primitive PI algebra is of the form M\_k(D) where D is the division algebra which is the commutant of the irreducible representation and that D is finite dimensional over its center. Note that k can be bounded in terms of the degree of the polynomial identity (in Passman's book you can get precise bounds on the degree) because the smallest degree polynomial identity satisfied by $M\_r(K)$ goes to infinity as $r$ goes to infinity.


If $G$ is countable, then every simple $\mathbb CG$-module is countable dimensional and hence the commutant of an irreducible representation has countable dimension over $\mathbb C$. But the only division algebra over $\mathbb C$ of countable dimension is $\mathbb C$ itself. Thus any irreducible representation of $\mathbb CG$ has image $M\_r(\mathbb C)$ where $r$ is bounded.


In any event once you know that each representation is of finite degree then you can take a normal abelian subgroup of index say $m$ and use Clifford theory to say each irreducible of $G$ has degree at most $m$.


