free group
A free group is the group with no surprises — the one obeying only the rules every group must. Pick some letters; form all strings you can make from them and their inverses; the only simplifications allowed are cancelling a letter against its own inverse standing right beside it. Nothing else is ever equal to the identity. It is the most ‘unconstrained’ group on its generators, every other group on those generators being a quotient of it.
Formally, the free group F(S) on a set S is the set of reduced words in S and S^(-1) (words with no adjacent pair xx^(-1) or x^(-1)x), under concatenation followed by free reduction. Its defining universal property: any function from S into a group G extends uniquely to a homomorphism F(S) to G. The cardinality of S is the rank, an isomorphism invariant; F of rank 1 is just Z, while ranks at least 2 are nonabelian. A presentation ⟨ S | ⟩ with no relations is exactly F(S).
Free groups are the building blocks of combinatorial group theory: every group is a quotient of a free group (just send a generating set to itself), which is what a presentation expresses. Geometrically the Cayley graph of a free group on a free basis is a tree, infinite and regular, with no cycles — the source of their negative-curvature behavior, exponential growth, and the fact that they act freely on trees.
The two matrices [1, 2; 0, 1] and [1, 0; 2, 1] in SL(2, Z) generate a free group of rank 2 (a classic ping-pong-lemma argument), embedding the abstract F_2 concretely inside 2×2 integer matrices.
Two simple integer matrices generate a free group of rank 2 inside SL(2, Z).
The rank-2 free group F_2 contains free subgroups of every countable rank, including countably infinite rank — a sharp contrast with vector spaces, where subspace dimension cannot exceed the ambient dimension.