the generators of a space group
A space group has infinitely many symmetry operations — because the lattice translations alone let you repeat forever in every direction. Yet you do not need an infinite list to describe it. The generators are a small, finite starter set of operations from which every other operation can be built by combining them, over and over. They are like a handful of basic dance moves that, chained together in different orders, produce the entire routine.
A typical generator set is just the three lattice translations (along a, b and c) plus a few point, screw or glide operations. You reach any operation in the group by composing generators and repeating them. Take P2_1/c: give the generators as the inversion centre at the origin and the 2_1 screw (from which the c-glide follows as their product), plus the lattice translations. Compose those and you generate all four general equivalent positions — and hence the whole space group — from almost nothing.
The International Tables print a chosen set of generators for every space group precisely so that crystallography software can build the full symmetry mechanically: feed in the generators, close the set under composition, and out come all the equivalent positions. The choice of generators is not unique — many different minimal sets generate the same group — but a small, standardized set is convenient and unambiguous once you also fix the origin and setting.
In P-1 a single generator beyond the lattice translations — the inversion at the origin — suffices: applying it once gives (-x, -y, -z), applying it twice returns you to (x, y, z), and combined with translations it reproduces the entire crystal's symmetry.
A few generators, closed under composition, unfold into a space group's infinite symmetry.
Generators are a description, not a physical thing in the crystal — the crystal has all its operations equally. Different textbooks or programs may list different generators for the same space group, and all are correct as long as they generate the same set.