# Ito-Michler theorem

From Groupprops

## Contents

## Statement

Suppose is a finite group and is a prime number. The following are equivalent:

- does not divide
*any*of the degrees of irreducible representations of over (or more generally, over some splitting field). - The -Sylow subgroup of is a Normal Sylow subgroup (?) and is also abelian.

Note that in the case that does not divide the order of at all, (2) is satisfied, so we do not need to assume that divides the order of . However, making that assumption does not weaken our theorem.

## Related notions

- Character degree graph of a finite group is an undirected graph associated with any finite group. Its vertex set is given precisely by the Ito-Michler theorem.

## Related facts

## Facts used

## Proof

### (2) implies (1)

This follows directly from fact (1), since the index of an abelian normal -Sylow subgroup is relatively prime to , hence all irreducible representations have degree dividing a number relatively prime to , forcing the degrees to be relatively prime to .

### (1) implies (2)

This is the hard part!