Degrees of Faithful Irreducible Representations of Certain Metabelian Groups and a Question of Sim

In this paper, we answer affirmatively a question of H S Sim on representations in characteristic $0$, for a class of metabelian groups. Moreover, we provide examples to point out that the analogous answer is no longer valid if the solvable group has derived length larger than 2. Let $F$ be a field of characteristic $0$ and $overline{F}$ be its algebraic closure. We prove that if $G$ is a finite metabelian group containing a maximal abelian normal subgroup which is a p-group with abelian quotient, all possible faithful irreducible representations over $F$ have the same degree and that the Schur index of any faithful irreducible $overline{F}$-representation with respect to $F$ is always $1$ or $2$. H S Sim had proven such a result for metacyclic groups when the characteristic of $F$ is positive and posed the question in characteristic $0$. Our result answers this question for the above class of metabelian groups affirmatively. We also determine explicitly the Wedderburn component corresponding to any faithful irreducible $overline{F}$-representation in the group algebra $F[G]$.