We prove, for certain pairs
G
,
G
′
G,G’
of finite groups of Lie type, that the
p
p
-fusion systems
F
p
(
G
)
\mathcal {F}_p(G)
and
F
p
(
G
′
)
\mathcal {F}_p(G’)
are equivalent. In other words, there is an isomorphism between a Sylow
p
p
-subgroup of
G
G
and one of
G
′
G’
which preserves
p
p
-fusion. This occurs, for example, when
G
=
G
(
q
)
G=\mathbb {G}(q)
and
G
′
=
G
(
q
′
)
G’=\mathbb {G}(q’)
for a simple Lie “type”
G
\mathbb {G}
, and
q
q
and
q
′
q’
are prime powers, both prime to
p
p
, which generate the same closed subgroup of
p
p
-adic units. Our proof uses homotopy-theoretic properties of the
p
p
-completed classifying spaces of
G
G
and
G
′
G’
, and we know of no purely algebraic proof of this result.