Isotypical-or-induced lemma

Statement with symbols
Let $$N$$ be a normal subgroup of a finite group $$G$$ and $$\rho$$ be an irreducible linear representation over any field $$k$$ (not necessarily algebraically closed, and not necessarily of characteristic zero). Then one of the following must hold:


 * There is a proper subgroup $$H$$ of $$G$$ containing $$N$$ such that $$\rho$$ is induced from an irreducible representation of $$H$$.
 * The restriction of $$\rho$$ to $$N$$ is isotypical: it is a direct sum of equivalent irreducible representations.

Related facts

 * Clifford's theorem

Corollaries

 * Degree of irreducible representation divides index of Abelian normal subgroup: This follows by combining the isotypical-or-induced lemma with the fact that degree of irreducible representation divides index of center.
 * Supersolvable implies monomial-representation: Every irreducible representation of a supersolvable group is monomial.

Breakdown for a non-normal subgroup
Let $$G$$ be the symmetric group on three elements and $$H$$ be a subgroup of order two. Then, $$H$$ is not a normal subgroup of $$G$$.

Let $$\rho$$ be an irreducible two-dimensional linear representation of $$G$$. Then:


 * The restriction of $$G$$ to $$H$$ is not isotypical: In fact, the restriction is the direct sum of the two irreducible representations of the cyclic group of order two.
 * The representation is not induced from any subgroup of $$G$$ containing $$H$$: Indeed, the degree of the representation is $$2$$, so it clearly cannot be induced from a subgroup containing $$H$$, which has index $$3$$.