[tzs]

Couldn't a large galaxy rotate as fast as a small galaxy, without the need for dark matter, if the large galaxy had a sufficiently more massive black hole at its center than the small galaxy.

The formula for the orbital period around a central mass is:

               3  1/2
           /  R  \
  T = 2 Pi |-----|
           \ G M /

where T is the period, R is radius of the orbit, G is the gravitational constant, and M is the mass of the central body. That's Newtonian, but I think that is good enough for this.

This says that if Big Galaxy has a radius k times that of Small Galaxy, it won't rotate slower than Small Galaxy as long as Big Galaxy's mass is at least k^3 times Small Galaxy's mass.

[salty_biscuits]

You are right that this could explain the velocity at a particular radius, but it's the radial velocity profile as you go out that hints at dark matter. The velocity as a function of radius is strange.