Isotypical-or-induced lemma

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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

Corollaries

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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