Clifford's theorem

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:

$$\langle \operatorname{Res}(\chi)_N^G, \mu \rangle \ne 0$$

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 $$[I_G(\mu):N]$$.

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 $$\mu$$.

Textbook references

 * , Page 70, Theorem 4.1 (formal statement, followed by proof).