For the sake of completeness, here is the answer to Question 1, part of which
is missing from the other answers:

**Proposition 1.** Let $G$ be a finite group. Consider the representations of
$G$ over $\mathbb{C}$. Let $\det G$ denote the determinant of the character
table of $G$. (Note that this is only defined up to sign, since the order of
the rows and of the columns of the character table can be chosen arbitrarily.)
Let $\operatorname*{perm}G$ denote the permanent of the character table of
$G$. Then, $\left(\det G\right)^2$ and $\operatorname*{perm}G$ are integers.

To prove this, we need the following lemmas:

**Lemma 2.** Let $G$ be a finite group. Then, there is a finite Galois field
extension $\mathbb{F}$ of $\mathbb{Q}$ such that all irreducible
representations of $G$ are defined over $\mathbb{F}$.

*Proof of Lemma 2.* There is a finite Galois field extension $\mathbb{K}$ of
$\mathbb{Q}$ such that all irreducible representations of $G$ are defined over
$\mathbb{K}$. Indeed, this is known as a *splitting field* of $G$; its
existence is part of Theorem 9.2.6 in Peter Webb, *A Course in Finite Group
Representation Theory*, 2016.

Consider this field extension $\mathbb{K}$. Let $\mathbb{F}$ be the Galois
closure of $\mathbb{K}$ over $\mathbb{Q}$ (or any other finite field extension
of $\mathbb{Q}$ that is Galois over $\mathbb{Q}$ and contains $\mathbb{K}$ as
a subfield). Then, all irreducible representations of $G$ are defined over
$\mathbb{F}$ (since they are defined over $\mathbb{K}$, but $\mathbb{F}$
contains $\mathbb{K}$ as a subfield). This proves Lemma 2. $\blacksquare$

**Lemma 3.** Let $G$ be a finite group. Let $\mathbb{F}$ be a field extension
of $\mathbb{Q}$ such that all irreducible representations of $G$
are defined over $\mathbb{F}$. Let $\chi:G\rightarrow\mathbb{F}$ be an
irreducible character of $G$. Let $\gamma:\mathbb{F}\rightarrow\mathbb{F}$ be
a $\mathbb{Q}$-algebra automorphism of $\mathbb{F}$. Then, $\gamma\circ
\chi:G\rightarrow\mathbb{F}$ is an irreducible character of $G$.

*Proof of Lemma 3.* This is a totally straightforward "isomorphisms preserve
all relative properties of objects they are applied to" argument, but for the
sake of completeness, let me spell it out (at least to some level of detail):

The map $\chi$ is an irreducible character of $G$, and thus is the character
of an irreducible representation $\rho$ of $G$. Consider this $\rho$, and WLOG
assume that $\rho$ is a representation over $\mathbb{F}$. (This can be assumed
since all irreducible representations of $G$ are defined over $\mathbb{F}$.)
Thus, $\rho$ is a group homomorphism from $G$ to $\operatorname*{GL}
\nolimits_{n}\left( \mathbb{F}\right) $ for some $n\geq1$. Consider this $n$.

The $\mathbb{Q}$-algebra automorphism $\gamma:\mathbb{F}\rightarrow\mathbb{F}$
induces a group automorphism $\widetilde{\gamma}:\operatorname*{GL}
\nolimits_{n}\left( \mathbb{F}\right) \rightarrow\operatorname*{GL}
\nolimits_{n}\left( \mathbb{F}\right) $ that transforms each matrix in
$\operatorname*{GL}\nolimits_{n}\left( \mathbb{F}\right) $ by applying
$\gamma$ to each entry of the matrix. The composition $\widetilde{\gamma}
\circ\rho:G\rightarrow\operatorname*{GL}\nolimits_{n}\left( \mathbb{F}
\right) $ is a group homomorphism (since $\widetilde{\gamma}$ and $\rho$ are
group homomorphisms), and thus is a representation of $G$. Moreover, the
character of this representation $\widetilde{\gamma}\circ\rho$ is $\gamma
\circ\chi$ (since $\operatorname*{Tr}\left( \widetilde{\gamma}\left(
A\right) \right) =\gamma\left( \operatorname*{Tr}A\right) $ for any matrix
$A\in\operatorname*{GL}\nolimits_{n}\left( \mathbb{F}\right) $). We shall
now show that this representation $\widetilde{\gamma}\circ\rho$ is irreducible.

Indeed, let $U$ be a subrepresentation of $\widetilde{\gamma}\circ\rho$ --
that is, an $\mathbb{F}$-vector subspace of $\mathbb{F}^{n}$ that is invariant
under the action of $\widetilde{\gamma}\circ\rho$. Consider the $\mathbb{Q}
$-module isomorphism $\overline{\gamma}:\mathbb{F}^{n}\rightarrow
\mathbb{F}^{n}$ that applies $\gamma$ to each coordinate of the vector. Since
$\gamma$ is a $\mathbb{Q}$-algebra homomorphism, we can easily see that
$\left( \widetilde{\gamma}\left( A\right) \right) \left( \overline
{\gamma}\left( v\right) \right) =\overline{\gamma}\left( Av\right) $ for
each $A\in\operatorname*{GL}\nolimits_{n}\left( \mathbb{F}\right) $ and each
$v\in\mathbb{F}^{n}$. Thus, we can easily see that $\overline{\gamma}
^{-1}\left( U\right) $ is an $\mathbb{F}$-vector subspace of $\mathbb{F}
^{n}$ that is invariant under the action of $\rho$ (since $U$ is an
$\mathbb{F}$-vector subspace of $\mathbb{F}^{n}$ that is invariant under the
action of $\widetilde{\gamma}\circ\rho$). In other words, $\overline{\gamma
}^{-1}\left( U\right) $ is a subrepresentation of $\rho$. Since $\rho$ is
irreducible, this entails that either $\overline{\gamma}^{-1}\left( U\right)
=0$ or $\overline{\gamma}^{-1}\left( U\right) =\mathbb{F}^{n}$. Since
$\overline{\gamma}$ is an isomorphism, we thus conclude that either $U=0$ or
$U=\mathbb{F}^{n}$.

Forget that we fixed $U$. We thus have shown that if $U$ is a
subrepresentation of $\widetilde{\gamma}\circ\rho$, then either $U=0$ or
$U=\mathbb{F}^{n}$. In other words, the representation $\widetilde{\gamma
}\circ\rho$ is irreducible (since its dimension is $n\geq1$). Thus, its
character is an irreducible character of $G$. In other words, $\gamma\circ
\chi$ is an irreducible character of $G$ (since $\gamma\circ\chi$ is the
character of $\widetilde{\gamma}\circ\rho$). This proves Lemma 3.
$\blacksquare$

*Proof of Proposition 1.* Lemma 2 shows that there is a finite Galois field
extension $\mathbb{F}$ of $\mathbb{Q}$ such that all irreducible
representations of $G$ are defined over $\mathbb{F}$. Consider this
$\mathbb{F}$. Let $\Gamma$ be the Galois group $\operatorname*{Gal}\left(
\mathbb{F}/\mathbb{Q}\right) $ (which consists of all $\mathbb{Q}$-algebra
automorphisms of $\mathbb{F}$). The Fundamental Theorem of Galois Theory shows
that the invariant ring $\mathbb{F}^{\Gamma}$ is $\mathbb{Q}$.

Let $\chi_{1},\chi_{2},\ldots,\chi_{r}$ be all irreducible characters of $G$
(listed without repetition). Note that these characters are maps from $G$ to
$\mathbb{F}$ (since all irreducible representations of $G$ are defined over
$\mathbb{F}$).

Let $c_{1},c_{2},\ldots,c_{r}$ be the conjugacy classes of $G$ (listed without repetition).

Let $\operatorname*{per}A$ denote the permanent of any square matrix $A$.

Let $C$ be the matrix $\left( \chi_{i}\left( c_{j}\right) \right) _{1\leq
i\leq r,\ 1\leq j\leq r}\in\mathbb{F}^{r\times r}$. This matrix $C$ is the
character table of $G$ (for the ordering of characters given by $\chi_{1}
,\chi_{2},\ldots,\chi_{r}$ and the ordering of conjugacy classes given by
$c_{1},c_{2},\ldots,c_{r}$). Thus, the definition of $\operatorname*{perm}G$
shows that $\operatorname*{perm}G$ is the permanent of $C$. In other words,
$\operatorname*{perm}G=\operatorname*{per}C$.

Let $\gamma\in\Gamma$. Thus, $\gamma$ is a $\mathbb{Q}$-algebra automorphism
of $\mathbb{F}$ (since $\gamma\in\Gamma=\operatorname*{Gal}\left(
\mathbb{F}/\mathbb{Q}\right) $). We shall show that $\gamma\left(
\operatorname*{perm}G\right) =\operatorname*{perm}G$.

The $\mathbb{Q}$-algebra automorphism $\gamma:\mathbb{F}\rightarrow\mathbb{F}$
induces a $\mathbb{Q}$-algebra automorphism $\widetilde{\gamma}:\mathbb{F}^{r\times
r}\rightarrow\mathbb{F}^{r\times r}$ that transforms each matrix in
$\mathbb{F}^{r\times r}$ by applying $\gamma$ to each entry of the matrix.

We define a map $f:\left\{ 1,2,\ldots,r\right\} \rightarrow\left\{
1,2,\ldots,r\right\} $ as follows:

Let $i\in\left\{ 1,2,\ldots,r\right\} $. Then, $\chi_{i}:G\rightarrow
\mathbb{F}$ is an irreducible character of $G$. Thus, Lemma 3 (applied to
$\chi=\chi_{i}$) shows that $\gamma\circ\chi_{i}:G\rightarrow\mathbb{F}$ is an
irreducible character of $G$. Hence, $\gamma\circ\chi_{i}=\chi_{j}$ for some
$j\in\left\{ 1,2,\ldots,r\right\} $ (since $\chi_{1},\chi_{2},\ldots
,\chi_{r}$ are all irreducible characters of $G$). This $j$ is uniquely
defined. We define $f\left( i\right) $ to be $j$.

Thus, we have defined a map $f:\left\{ 1,2,\ldots,r\right\} \rightarrow
\left\{ 1,2,\ldots,r\right\} $ with the property that
\begin{equation}
\gamma\circ\chi_{i}=\chi_{f\left( i\right) }\qquad\text{for each }
i\in\left\{ 1,2,\ldots,r\right\} .
\label{eq.darij1.1}
\tag{1}
\end{equation}

If two distinct elements $i$ and $j$ of $\left\{ 1,2,\ldots,r\right\} $
would satisfy $f\left( i\right) =f\left( j\right) $, then they would
satisfy $\gamma\circ\chi_{i}=\gamma\circ\chi_{j}$ (by \eqref{eq.darij1.1}
) and therefore $\chi_{i}=\chi_{j}$ (since $\gamma$ is invertible), which
would contradict the fact that $\chi_{1},\chi_{2},\ldots,\chi_{r}$ are
distinct. Thus, two distinct elements $i$ and $j$ of $\left\{ 1,2,\ldots
,r\right\} $ always satisfy $f\left( i\right) \neq f\left( j\right) $. In
other words, the map $f$ is injective. Hence, $f$ is a permutation (since $f$
is an injective map from $\left\{ 1,2,\ldots,r\right\} $ to $\left\{
1,2,\ldots,r\right\} $).

Now, from $C=\left( \chi_{i}\left( c_{j}\right) \right) _{1\leq i\leq
r,\ 1\leq j\leq r}$, we obtain
\begin{align*}
\widetilde{\gamma}\left( C\right) =\left( \gamma\left( \chi_{i}\left(
c_{j}\right) \right) \right) _{1\leq i\leq r,\ 1\leq j\leq r}=\left(
\chi_{f\left( i\right) }\left( c_{j}\right) \right) _{1\leq i\leq
r,\ 1\leq j\leq r},
\end{align*}
since each $i,j\in\left\{ 1,2,\ldots,r\right\} $ satisfy
\begin{align*}
\gamma\left( \chi_{i}\left( c_{j}\right) \right) =\left( \gamma\circ
\chi_{i}\right) \left( c_{j}\right) =\chi_{f\left( i\right) }\left(
c_{j}\right) \qquad\left( \text{by \eqref{eq.darij1.1}}\right) .
\end{align*}
Thus, the matrix $\widetilde{\gamma}\left( C\right) $ is obtained from the
matrix $\left( \chi_{i}\left( c_{j}\right) \right) _{1\leq i\leq r,\ 1\leq
j\leq r}$ by permuting the rows (since $f:\left\{ 1,2,\ldots,r\right\}
\rightarrow\left\{ 1,2,\ldots,r\right\} $ is a permutation). In other words,
the matrix $\widetilde{\gamma}\left( C\right) $ is obtained from the matrix
$C$ by permuting the rows (since $C=\left( \chi_{i}\left( c_{j}\right)
\right) _{1\leq i\leq r,\ 1\leq j\leq r}$). Hence, $\operatorname*{per}
\left( \widetilde{\gamma}\left( C\right) \right) =\operatorname*{per}C$
(since the permanent of a matrix does not change when its rows are permuted).

But the definition of $\widetilde{\gamma}$ yields that $\operatorname*{per}
\left( \widetilde{\gamma}\left( C\right) \right) =\gamma\left(
\operatorname*{per}C\right) $ (since $\gamma$ is a $\mathbb{Q}$-algebra
homomorphism). Hence, $\gamma\left( \operatorname*{per}C\right)
=\operatorname*{per}\left( \widetilde{\gamma}\left( C\right) \right)
=\operatorname*{per}C$. In view of $\operatorname*{perm}G=\operatorname*{per}
C$, this rewrites as $\gamma\left( \operatorname*{perm}G\right)
=\operatorname*{perm}G$.

Forget that we fixed $\gamma$. We thus have shown that $\gamma\left(
\operatorname*{perm}G\right) =\operatorname*{perm}G$ for each $\gamma
\in\Gamma$. In other words, $\operatorname*{perm}G$ belongs to the invariant
ring $\mathbb{F}^{\Gamma}$. In other words, $\operatorname*{perm}G$ belongs to
$\mathbb{Q}$ (since the invariant ring $\mathbb{F}^{\Gamma}$ is $\mathbb{Q}$).

But all values of the characters $\chi_{1},\chi_{2},\ldots,\chi_{r}$ are sums
of roots of unity (because they are traces of matrices $A\in\operatorname*{GL}
\nolimits_{n}\left( \mathbb{F}\right) $ that satisfy $A^{\left\vert
G\right\vert }=I_{n}$, and the eigenvalues of such a matrix are roots of
unity), and thus are algebraic integers. Hence, all entries of the matrix $C$
are algebraic integers (since all these entries are values of the characters
$\chi_{1},\chi_{2},\ldots,\chi_{r}$). Thus, the permanent $\operatorname*{per}
C$ of this matrix $C$ is an algebraic integer (since the algebraic integers
form a ring). In other words, $\operatorname*{perm}G$ is an algebraic integer
(since $\operatorname*{perm}G=\operatorname*{per}C$). Hence,
$\operatorname*{perm}G$ is an algebraic integer in $\mathbb{Q}$ (since
$\operatorname*{perm}G$ belongs to $\mathbb{Q}$). Since the only algebraic
integers in $\mathbb{Q}$ are integers (because the ring $\mathbb{Q}$ is
integrally closed), this entails that $\operatorname*{perm}G$ is an integer.

A similar argument shows that $\left( \det G\right) ^{2}$ is an integer.
(Here we need to use the fact that the square of the determinant of a matrix
does not change when its rows are permuted. This is because the determinant
gets multiplied by a power of $-1$.) Thus, Proposition 1 is proved.
$\blacksquare$

Permfectis a great term. $\endgroup$5more comments