Clifford's theorem

This fact is related to: linear representation theory
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Statement

Verbal statement

The restriction of any irreducible complex character of a group, to a normal subgroup, is a multiple of the sum of all conjugates in the whole group of some irreducible character of the normal subgroup.

Statement with symbols, using character-theoretic language

Let $G$ be a finite group and $N$ a normal subgroup of $G$. Let $\chi$ be a complex irreducible character of $G$ and $\mu$ of $N$ such that:

$\langle \operatorname {Res} (\chi )_{N}^{G},\mu \rangle \neq 0$

Then:

$\operatorname {Res} (\chi )_{N}^{G}=a\left(\sum _{i=1}^{t}\mu ^{(g_{i})}\right)$

where $g_{i}\in G$ and $\mu ^{(g)}$ denotes the character:

$n\mapsto \mu (gng^{-1})$

Further $a$ and $t$ are positive integers dividing the index $[G:N]$. In fact, $t$ is the index of the subgroup $I_{G}(\mu )$, defined as:

$\left\{g\in G|\mu ^{(g)}=\mu \right\}$

$I_{G}(\mu )$ is termed the inertial subgroup.

Further, $e$ divides the index $[I_{G}(\mu ):N]$.

Statement with symbols, using module-theoretic language

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Particular cases

Conjugacy-closed subgroups

A conjugacy-closed subgroup is a subgroup such that any two elements of the subgroup conjugate in the whole group, are also conjugate in the subgroup. If $N$ is conjugacy-closed, then $I_{G}(\mu )=G$ for any $\mu$ and thus, in this case, the restriction of the irreducible character from $G$ to $N$ is simply a multiple of $\mu$.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 70, Theorem 4.1 (formal statement, followed by proof).