Isotypical-or-induced lemma

This fact is related to: linear representation theory
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Statement

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

Further information: symmetric group:S3, linear representation theory of symmetric group:S3

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 $3$ .

Proof

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