# Isotypical-or-induced lemma

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This fact is related to: linear representation theory

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## Contents

## Statement

### Statement with symbols

Let be a normal subgroup of a finite group and be an irreducible linear representation over any field (not necessarily algebraically closed, and not necessarily of characteristic zero). Then one of the following must hold:

- There is a proper subgroup of containing such that is induced from an irreducible representation of .
- The restriction of to is isotypical: it is a direct sum of equivalent irreducible representations.

## Related facts

### 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 be the symmetric group on three elements and be a subgroup of order two. Then, is *not* a normal subgroup of .

Let be an irreducible two-dimensional linear representation of . Then:

- The restriction of to 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 containing : Indeed, the degree of the representation is , so it clearly cannot be induced from a subgroup containing , which has index .

## Proof

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