Exercise 14, § 21

A representation of G is called monomial if it is the direct sum of representations induced by degree 1 representations of subgroups of G.

a) Suppose that the representations of the proper subgroups of G are monomial and that G contains a noncentral commutative normal subgroup H. Prove that every faithful irreducible representation of G is monomial (observe that the restriction to H of such a representation is not isotypical, and apply Exercise 9, a)).

b) Deduce that all representations of supersolvable groups (I, §6, p. 146, Exercise 26), and in particular of nilpotent groups, are monomial.

c) Take G to be the group $\mathbf{S}\mathbf{L}(2,\mathbf{F}_3)$. Its center Z has order 2, and the group $G/Z$ is isomorphic to the alternating group $\mathfrak{A}_4$ (II, §10, p. 422, Exercise 14, g)); in particular, G is solvable. Prove that its derived group has index 3 (cf. I, §5, p. 142, Exercise 10). Using formula (8) of VIII, p. 407, deduce that G admits an irreducible representation of degree 2, and prove that this representation is not monomial.