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Recall that

We say an extension field $E$ of $F$ is a splitting field of $A$ if every simple $E{\otimes }_{F}A$-module is absolutely irreducible.

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Definition

If $k$ is splitting for the group algebra $kG$, then we say that $k$ is splitting for $G$.

Brauer

If $k$ contains an $\exp (G)$-th root of unity, then $k$ is splitting for $G$.

Remark. 大概意思是说Schur lemma中代数闭域的条件可以放弱，最弱能到这个分裂域。

Recall that if $\mathrm{char}k\mid |G|$, then $kG$ is not semisimple.

Maschke theorem

If $\mathrm{char}k\nmid |G|$, then $kG$ is semisimple.

\begin{proof} See ^mgo6js. \end{proof}

Corollary

Let $\mathrm{char}k\nmid |G|$. Then any finitely generated $k$-representation of $G$ is a sum of irreducible representations of $G$.

Remark. 讲了两个名词：

Characteristic Theory

Proposition

Let $G$ be a finite group, and let $k$ be a field with $\mathrm{char}k\nmid |G|$.

• As a ring, $kG$ is a direct sum of matrix algebras over division rings.
• Suppose in addition $k$ is splitting for $G$. Let $S_1,\cdots ,S_r$ be pairwise non-isomorphic simple $kG$-module, and let $d_i={\dim }_k{S}_i$. Then $d_i$ equals the multiplicity with which $S_i$ is a summand of the regular representation of $G$, and $|G|⩾d_1^2+\cdots +d_r^2$ with equality iff $S_1,\cdots ,S_r$ is a complete set of representative of isomorphisms of simple $kG$-modules.

然后就开始讲$\mathbb{C}$上的特征标，一节课讲到了这里：

The irreducible characters of $G$ form an orthonormal basis with respect to inner product introduced above.

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Recall that $Z(\mathbb{Z}G)={\oplus }_{C\in \mathrm{Cl}(G)}\mathbb{Z}(SC)$.

Proposition

The center $Z(\mathbb{Z}G)$ is integral over $\mathbb{Z}$. Hence if $C_1,\cdots ,C_r$ are conjugacy classes of $G$ and ${\lambda }_1,\cdots ,{\lambda }_r$ are algebraic integers in $\mathbb{C}$, then ${\sum }_{i=1}^r{\lambda }_i(S{C}_i)$ is integral over $\mathbb{Z}$.

\begin{proof} By ^jy6o7s. In fact, we can prove $\mathbb{Z}G$ is integral over $\mathbb{Z}$. \end{proof}