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.

Symbolic statement

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:

$⟨\mathrm{R}\mathrm{e}\mathrm{s}(\chi {)}_N^G,\mu ⟩\ne 0$

Then:

$\mathrm{R}\mathrm{e}\mathrm{s}(\chi {)}_N^G=a(\sum _{i=1}^t{\mu }^{(g_i)})$

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

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

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:

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

$I_G(\mu )$ is termed the inertial subgroup.

Further, $e$ divides the index $[I_G(\mu ):N]$.

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 $\mu$.