A normal version of Brauer's height zero conjecture

The celebrated It\ᵒ-Michler theorem asserts that a prime p does not divide the degree of any irreducible character of a finite group G if and only if G has a normal and abelian Sylow p-subgroup. The principal block case of the recently-proven Brauer's height zero conjecture isolates the abelian part in the It\ᵒ-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture.