4. Prove that if $G$ is nontrivial, then every irreducible $FG$-module has degree $< |G|$.
18. Prove that if $φ : G → GL_n(ℂ)$ 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 $W⊆C_V(g)$, where $C_V(g)$ is the set of elements of $V$ fixed by $g$.
(b) Prove that $V$ is generated as an additive group by the sets $C_V(g)$ 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 $\sum_{g∈G} gv$ 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 $ℂ^n → ℂ^n$. 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 $ℂ^n$, 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 $ℂ^n$ is that of scalar multiplication. If we decompose $ℂ^n$ 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 $Z(G/\text{ker }φ)$ 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 $\text{ker }φ$ 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 $W_i$ are contained in $C_V(g_i)$ for some nonidentity $g_i∈G$, by part (a). It follows $V$ is generated as an abelian group by the $C_V(g_i)$.
(22) We may assume a representation $φ$ of $P$ over $F$ is faithful by passing to the quotient $P / \text{ker }φ$ 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 $\text{det }φ(x)-1 = 0$. As $x$ commutes with every element of $P$, we have $\text{ker }φ(x)-1$ 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 $N \unlhd G$ 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 $ng·w = gg^{-1}ng·w = g(g^{-1}ng)·w = g·w$, 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 $α_{xN} = \sum_{g∈xN} g ∈ FG$. It is clear, then, that $α_{xN}∈W$, the $α_{xN}$ 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 $α_{xN}$, so the $α_{xN}$ form an $F$-basis for $W$. Define an $F$-linear isomorphism $φ : W→F(G/N)$ given by $φ(α_{xN}) = \overline{x}$. This is in fact an $F(G/N)$-module isomorphism, since $φ(\overline{g}α_{xN}) = φ(α_{gxN}) = \overline{gx} = \overline{g}φ(α_{xN})$.$~\square$
We proceed by induction on the order of $P$. Suppose $FP = V_1 \oplus V_2$ as $FP$-modules. If $P$ is nontrivial, let $x∈Z(P)$ be of order $p$. Then on any $FP$-submodule of $FP$, we have $(x-1)^p = x^p - 1 = 0$ as $FP$-module transformations, hence $x-1$ has nontrivial kernel in each of $V_1, V_2$; this is to say $\langle~x~\rangle$ fixes nontrivial elements in each of $V_1,V_2$, say $W_1,W_2$ respectively. Then if $W$ is the $FP$-submodule of elements of $FP$ fixed by $\langle~x~\rangle$, we in fact have $W = W_1 \oplus W_2$. This nontrivial direct sum expression as $FP$-modules translates to the same as $F(P/\langle~x~\rangle)$-modules. But now $W ≅ F(P/\langle~x~\rangle)$ may be written as a nontrivial direct sum, a contradiction.$~\square$
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