This answer is spurred a little by @claude chuber's comment. I appreciate his clarifications of my comments, and his questioning of them as well.
As he states: the OPs statement from a book, 'space is not expanding within galaxies but rather within galaxies' "is wrong"
First, a description and explanation of what happens to the cosmological (what the OP calls space) expansion. The simplest description is that yes, there is cosmological expansion everywhere, inside galaxies, in between them in a cluster of galaxies, inside planets and the solar system, inside atoms and molecules, etc. But the expansion is a gravitational effect, and it can be counteracted by other forces - such as the gravitation due to the other bodies near them, electrical forces, nuclear forces, etc. And it turns out that for all those cases mentioned above (solar system, galaxies, clusters, atoms, planets, etc), except the forces between galaxy clusters, those other forces are stronger than the gravitational effect that causes the cosmological expansion. So, the cosmological expansion in those cases is so small in comparison as to be negligible, or if not negligible, makes a very small contributions. For expansion between galaxy clusters and any larger sizes the other forces have little effect, and the expansion proceeds and is measurable.
The reason why the expansion is more noticeable and effective for longer distances is that the expansion rate (i.e., velocity of recession between two objects at these large distances) increases with distance. That is what Hubble discovered, recession velocity is linearly proportional to distance. Remember that in the cosmological expansion everything is going away from everything else (imagine any two dots painted on the surface of an inflating balloon, as the ballon gets inflated they separate more and more), and the further away they are the faster they separate even more.
But two atoms, or two planets in the solar system, or two stars in a galaxy are not far enough from each other, and the expansion would be small. But it is even worse: their electrical and magnetic interactions (for atoms) and gravitational attractions (for the solar system, planets and galaxies) with each other are strong enough that the weak inflation effect is counteracted. They are called bound systems, and inflation does not affect them. So, for instance, for our galaxy, the Milky Way, and the Andromeda Galaxy, are about 1 Mpsec (about 3 light years)(number is approximate, could be off some) apart. Based on Hubble's constant of 67-70 Kms/sec/Mpsec they should receding from each other at about 67-70 Kms/sec. But what is measured is that they are going towards each other at a speed of about 119 Kms/sec. Their massive gravitational attraction is overwhelming the cosmological expansion at those distances. We will collide with the Andromeda Galaxy in about 3 billion years.
The gravitational effect at any small region of spacetime is the combination of many effects (for many cases one can treat one as a perturbation on the main effect, but if they are all about the same strength one has to treat them all together and it is more than we can do now. Now we treat them as perturbations on each other). The nearby objects have a large effect relative to each other, but on a large scale, call it cosmological scales, that whole group of galaxies that form our local cluster, are receding away from everything else due to the expansion. It is on scales of maybe 10 Mpsec or so, and certainly at the 30-50 Mpsec range, where the inflation takes over and is measurable. The linear relation of speed of recession to distance is good up to distances of a few billion light years, then some nonlinearities enter in.
So, yes, there is cosmological expansion at all levels, but it is very small and overwhelmed by local forces at scales less than about 10-30 Mpsec.
It is also worthwhile noting that the cosmological expansion is true, exactly as estimated and modeled, ON THE AVERAGE. There are irregularities at smaller scales, where the matter and energy density is higher, and other areas where it is less. At the 100 Mpsec scale the universe looks homogeneous and isotropic. But those over densities and under densities, make local groups of stars and galaxies have some peculiar velocities with respect to the overall expansion. From the earth, when we subtract our overall peculiar velocity in our measurements, we see, for instance, the cosmic background background (CMB), as homogeneous and isotropic (except for the remnants of the Planckian quantum super-microscopic perturbations right after the Big Bang that led later, as the universe evolved, to the galaxies and stars) - we've measured that on the CMB and it is very consistent with the predictions of the cosmological model.
THe cosmological model acceptEd today is the Lambda CDM model, with parameters best taken form the latest Planck satellite/collaboration data release.
See for references
Universe expansion, and small scale effects
The Hubble law and constant. See the numbers from Wikipedia, at https://en.m.wikipedia.org/wiki/Hubble's_law, mostly 67 to 72.
See the numbers from the Planck data release, it shows less variation when averaged with various measurements. See it at https://en.m.wikipedia.org/wiki/Planck_(spacecraft)#2015_data_release