Square of degree of irreducible representation divides order of inner automorphism group in finite nilpotent group

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Statement

In characteristic zero

Suppose G is a Finite nilpotent group (?) and K is a splitting field for G of characteristic zero. Then, for any irreducible linear representation of G over K, the square of the degree divides the order of the inner automorphism group of G, which is the same as the index of the center of G.

General case

Since degrees of irreducible representations are the same for all splitting fields, the truth of the statement for splitting fields in characteristic zero implies its truth for splitting fields in any characteristic not dividing the order of the group.

Related facts

Similar facts

Opposite facts

Facts used

  1. Equivalence of definitions of finite nilpotent group (reduce to the case where G has prime power order)
  2. Order of inner automorphism group bounds square of degree of irreducible representation
  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.