Clifford Theorem

Theorem

Let $k$ be any field, $U$ a simple $kG$ module and $N$ a normal subgroup of $G$. We may write $U{\downarrow }_N^G=S_1^{a_1}\oplus \cdots \oplus S_r^{a_r}$ where the $S_i$ are nonisomorphic simple $kN$-modules, occurring with multiplicities $a_i$. (We refer to the summands $S_i^{a_i}$ as the homogeneous components.) Then

• $G$ permutes the homogeneous components transitively;
• $a_1=a_2=\cdots =a_r$ and $\dim S_1=\dim S_2=\cdots =\dim S_r$; and
• if $H={\mathrm{Stab}}_G\left(S_1^{a_1}\right)$ then $U\cong S_1^{a_1}{\uparrow }_H^G$ as $kG$-modules.

We learnt it before:

• if ${\psi }_1,\cdots ,{\psi }_m$ are the constituents of $\chi \downarrow H$, then $\chi \downarrow H=e({\psi }_1+\cdots +{\psi }_m)$for some positive integer $e$.

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