  We call a pro-$p$ group $G$ Frattini-injective if distinct finitely generated
subgroups of $G$ have distinct Frattinis. This paper is an initial effort
toward a systematic study of Frattini-injective pro-$p$ groups (and several
other related concepts). Most notably, we classify the $p$-adic analytic and
the solvable Frattini-injective pro-$p$ groups, and we describe the lattice of
normal abelian subgroups of a Frattini-injective pro-$p$ group.
  We prove that every maximal pro-$p$ Galois group of a field that contains a
primitive $p$th root of unity (and also contains $\sqrt{-1}$ if $p=2$) is
Frattini-injective. In addition, we show that many substantial results on
maximal pro-$p$ Galois groups are in fact consequences of Frattini-injectivity.
For instance, a $p$-adic analytic or solvable pro-$p$ group is
Frattini-injective if and only if it can be realized as a maximal pro-$p$
Galois group of a field that contains a primitive $p$th root of unity (and also
contains $\sqrt{-1}$ if $p=2$); and every Frattini-injective pro-$p$ group
contains a unique maximal abelian normal subgroup.
