Degree of irreducible representation divides index of center
This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group
This article states a result of the form that one natural number divides another. Specifically, the (degree of a linear representation) of a/an/the (irreducible linear representation) divides the (index of a subgroup) of a/an/the (center).
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Let be a finite group and be an algebraically closed field of characteristic zero. Let be an irreducible representation of over . Then, the degree of divides the index of the center in . (Note that since the quotient by the center equals the inner automorphism group, this is equivalent to saying that the degree of divides the order of the inner automorphism group).
Similar facts about degrees of irreducible representations
- Degree of irreducible representation divides group order
- Order of inner automorphism group bounds square of degree of irreducible representation
- Degree of irreducible representation divides index of abelian normal subgroup
- Sum of squares of degrees of irreducible representations equals order of group