Degree of irreducible representation divides index of abelian subgroup in finite nilpotent group

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Statement

Suppose G is a finite nilpotent group, K is a splitting field for G of characteristic zero, and H is an abelian subgroup of G. Then, for any irreducible linear representation of G over K, the degree of the representation divides the index [G:H].

This gives a divisibility constraint on the degrees of irreducible representations.

Related facts

Similar facts

In the case of a finite nilpotent group, the statement that the degree of any irreducible representation divides the index of any abelian subgroup is prima facie stronger. In practice, for most small orders, the two statements have the same power, due to the Jonah-Konvisser abelian-to-normal replacement theorem which guarantees that, under certain conditions, any abelian subgroup of a given order can be replaced by an abelian normal subgroup of the same order.

Opposite facts

Facts used

  1. Equivalence of definitions of finite nilpotent group
  2. Degree of irreducible representation is bounded by index of abelian subgroup
  3. Degree of irreducible representation divides order of group

Proof

The proof has two steps:

  1. Reduce to the case where G is a group of prime power order using Fact (1).
  2. Now use Facts (2) and (3) and the observation that for two numbers that are powers of the same prime, the smaller one divides the bigger one.