general linear group (K-theory)
In K-theory one rarely cares about a matrix of a fixed size; what matters are properties that persist when you pad a matrix with extra 1's down the diagonal. The infinite general linear group is the device that makes this padding official: it is the union of all the finite general linear groups, glued together so that an n-by-n matrix and the same matrix bordered to (n+1)-by-(n+1) are simply the same element. Stabilizing in this way smooths out the low-rank accidents that obscure the deeper invariants.
Formally, for a ring R one forms the directed system GL(1, R) -> GL(2, R) -> GL(3, R) -> ... where GL(n, R) -> GL(n+1, R) sends A to the block matrix [A, 0; 0, 1]. The infinite general linear group GL(R) is the direct limit (colimit) of this system, equivalently the group of invertible infinite matrices that differ from the identity in only finitely many entries. Every element lives in some GL(n, R), and two elements are equal in GL(R) iff they agree after stabilizing into a common GL(N, R).
GL(R) is the workhorse of higher K-theory. Its abelianization is K_1(R); its commutator subgroup is the elementary group E(R); and applying Quillen's plus construction to the classifying space BGL(R) produces a space whose homotopy groups are the higher K-groups K_n(R) for n at least 1. Thus essentially all of K-theory above K_0 is encoded in the homotopy type of this single, stabilized linear group.
Take R a field F. An element of GL(F) is an invertible matrix of some finite size with entries in F, regarded up to adding identity blocks. Its image in the abelianization K_1(F) = F^* is its determinant; the determinant is well-defined precisely because det[A, 0; 0, 1] = det A.
GL(F) is the rising union of GL(n, F), padded by identity blocks.
Stabilization is essential because the unstable groups GL(n, R) have homology that only settles down as n grows; the homology of the limit GL(R) is the homological stability range made permanent. Quillen's plus construction is precisely the tool that converts this group homology into homotopy groups without changing it.