p ‐parts of character degrees

We show that if p is an odd prime and G is a finite group satisfying the condition that p 2 divides the degree of no irreducible character of G , then| G : O p( G ) | p ⩽ p 4 , whereO p ( G )is the largest normal p ‐subgroup of G , and if P is a Sylow p ‐subgroup of G , then P ' 'is subnormal in G . Our investigations suggest that if p a is the largest power of p dividing the degrees of irreducible characters of G , then| G : O p( G ) | pis bounded byp f ( a ) , where f ( a ) is a function in a and P ( a + 1 )is subnormal in G .