4. Prove that if G is nontrivial, then every irreducible FG-module has degree <|G|.
18. Prove that if φ:G→GLn(ℂ) is an irreducible matrix representation and A is an n×n matrix commuting with φ(g) for all g∈G, then A is a scalar matrix. Deduce that if φ is a faithful, irreducible, complex representation then the center of G is cyclic and φ(g) is a scalar matrix for all elements z in the center of G.
19. Prove that if G is an abelian group then any finite dimensional complex representation of G is equivalent to a representation into diagonal matrices.
21. Let G be a group with noncyclic center acting on a finite-dimensional vector space V over F, where F is a field of characteristic p not dividing the order of G.
(a) Prove that if W is an irreducible FG-submodule of V then there is some nonidentity g∈G such that W⊆CV(g), where CV(g) is the set of elements of V fixed by g.
(b) Prove that V is generated as an additive group by the sets CV(g) as g runs over all nonidentity elements of G.
22. Let p be a prime, let P be a p-group and let F be a field of characteristic p. Prove that the only irreducible representation of P over F is the trivial representation.
24. Let p be a prime, let P be a nontrivial p-group and let F be a field of characteristic p. Prove that the regular representation is indecomposable.
Proof: (4) Let V be an FG-module, and let v∈V be nonzero. Then either {gv | g∈G} are linearly dependent over F, in which case they generate a G-stable subspace of dimension strictly less than |G|, or they are linearly independent, in which case the nonzero element ∑g∈Ggv generates a one-dimensional line that is fixed by each g∈G. As such, we see that any FG-module not containing any proper nonzero FG-submodule must be of dimension <|G|.
(18) Let λ be an eigenvalue of A, i.e. A−λ has nontrivial kernel as a linear transformation ℂn→ℂn. Since A (and λ) commute with every element of φ(G), we see the kernel of A−λ is a G-submodule. Since the representation is irreducible, this is to say the kernel is all of ℂn, so that A−λ=0 and A is a scalar matrix. When φ is faithful, this implies the center of G embeds into the group of nonzero scalar matrices, i.e. ℂ×, so that the center of G is cyclic.
(19) By the above exercise, we find that the action of G on any irreducible G-stable subspace of ℂn is that of scalar multiplication. If we decompose ℂn into a direct sum of irreducible ℂG-submodules and observe the matrix representation of G with respect to the ℂ-basis of this decomposition, we find that they are diagonalized matrices.
(21)(a) This is a consequence fact that for any irreducible representation φ of a group G, we find Z(G/ker φ) is cyclic (cf. exercise 14d). For if W is an irreducible FG-submodule, this offers an irreducible representation of G (by the action on W). It follows that if Z(G) is not cyclic, then ker φ is nontrivial, i.e. there exists a nonidentity element of G that fixes all of W.
(b) Decompose V into a finite direct sum of irreducible FG-submodules by Maschke's theorem. Each of these summands Wi are contained in CV(gi) for some nonidentity gi∈G, by part (a). It follows V is generated as an abelian group by the CV(gi).
(22) We may assume a representation φ of P over F is faithful by passing to the quotient P/ker φ if necessary. But if P is a nontrivial p-group, it has a nontrivial center, and there is thus an element x∈P of order p commuting with every element in P. If FP is acting irreducibly on a vector space F over V via the representation φ, then V is finite dimensional, and as finite-dimensional matrices we have φ(x)p−1=(φ(x)−1)p=0, thus det φ(x)−1=0. As x commutes with every element of P, we have ker φ(x)−1 determines a P-stable subspace of V, hence is all of V, hence x=1, a contradiction since φ is faithful.
(24) Lemma: Let G be a group (not necessarily finite), let F be a field, and let V be an FG-module. If N⊴G is a normal subgroup, then the set of elements W⊆V that are fixed by N is an FG-submodule, and there is a natural F(G/N)-module action on W. The FG-submodules and F(G/N)-submodules of W are the same. Furthermore, if the module V is itself FG affording the regular representation on G, then the induced module is isomorphic to the regular representation of G/N.
Proof: It is clear W is a subspace, and if n∈N, g∈G, and w∈W, then ng·w=gg−1ng·w=g(g−1ng)·w=g·w, so that W is G-stable. As well, note that the module structure of FG on V is given by the action of G on V; since W is an G-stable, this restricts to an action of G on W; since N fixes each element of W, this induces a natural action of G/N on W, thereby turning W into a F(G/N)-module under this action. We see an F-subspace is G-stable if and only if it is G/N-stable, so the FG-submodules and F(G/N)-submodules are the same.
Suppose V is the module given by the regular representation, i.e. V=FG and the module action is given by multiplication. For each distinct left coset xN define αxN=∑g∈xNg∈FG. It is clear, then, that αxN∈W, the αxN are linearly independent over F, and that any element of FG fixed by each element of N must retain equivalent coefficients on elements in the same left coset of N, hence be an F-linear combination of the αxN, so the αxN form an F-basis for W. Define an F-linear isomorphism φ:W→F(G/N) given by φ(αxN)=¯x. This is in fact an F(G/N)-module isomorphism, since φ(¯gαxN)=φ(αgxN)=¯gx=¯gφ(αxN). ◻
We proceed by induction on the order of P. Suppose FP=V1⊕V2 as FP-modules. If P is nontrivial, let x∈Z(P) be of order p. Then on any FP-submodule of FP, we have (x−1)p=xp−1=0 as FP-module transformations, hence x−1 has nontrivial kernel in each of V1,V2; this is to say ⟨ x ⟩ fixes nontrivial elements in each of V1,V2, say W1,W2 respectively. Then if W is the FP-submodule of elements of FP fixed by ⟨ x ⟩, we in fact have W=W1⊕W2. This nontrivial direct sum expression as FP-modules translates to the same as F(P/⟨ x ⟩)-modules. But now W≅F(P/⟨ x ⟩) may be written as a nontrivial direct sum, a contradiction. ◻
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