Bounds on the order of cross characteristic subgroups of the finite simple groups of Lie type

Let X(r) and G(q) be finite groups of Lie type and r and q be coprime. If G(q) is embedded in X(r), then the Landazuri-Seitz-Zalesskii theorem implies that G(q) is small relative to X(r). We formalize this observation and illustrate how it can be used with some applications.