Nielsen-Schreier theorem
Take a free group — a group with no relations — and look inside it at any subgroup, however weirdly chosen. The Nielsen-Schreier theorem promises something clean: that subgroup is again free. Freeness is hereditary. There are no hidden relations lurking in subgroups; the absence of relations passes intact from a free group down to all of its subgroups.
Precisely: every subgroup H of a free group F is itself a free group. The Schreier index formula sharpens this when the index [F : H] = n is finite and F has finite rank r: then H has rank 1 + n(r - 1). So unless n = 1 or r = 1, a finite-index subgroup of a free group has strictly larger rank than the whole group — subgroups can be ‘bigger’ in rank, a phenomenon impossible for abelian groups.
The cleanest proof is topological and is the prototype of geometric group theory. A free group of rank r is the fundamental group of a wedge of r circles; a subgroup H corresponds to a covering space of that wedge, which is a graph; and the fundamental group of any graph is free, with rank equal to one minus its Euler characteristic. Nielsen and Schreier's original combinatorial proofs (via Nielsen transformations and Schreier transversals) predate this viewpoint but compute the same answer.
In F_2 = ⟨ a, b ⟩, the commutator subgroup (index infinite, the kernel of the map to Z × Z) is free of infinite rank, with basis the commutators [a^m, b^n]. And the index-2 subgroup of even-length words is free of rank 1 + 2(2 - 1) = 3.
An index-2 subgroup of F_2 is free of rank 3 — more generators than its parent.
The full theorem for subgroups of arbitrary (possibly infinite) rank requires the axiom of choice; it is in fact equivalent to a weak choice principle over Zermelo-Fraenkel set theory.