  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.
