special linear group SL(n)

Among all invertible matrices, single out those whose determinant is exactly 1. These are the volume-and-orientation-preserving maps: they can shear and stretch space wildly, but the total signed volume of any region comes out unchanged. That subgroup is the special linear group SL(n).

Formally SL(n, F) = { A in GL(n, F) : det(A) = 1 }. It is a group because det is multiplicative: det(AB) = det(A)det(B) = 1 and det(A^-1) = 1/det(A) = 1. In fact SL(n) is precisely the KERNEL of the group homomorphism det : GL(n, F) -> F^* (the nonzero scalars under multiplication), which instantly makes it a normal subgroup.

Geometrically, det = 1 means a unit cube maps to a solid of the same volume with the same orientation — no flipping, no scaling of volume. Contrast SO(n), which preserves the much stronger structure of lengths and angles; SL(n) only protects volume, so it is far bigger and includes shears that distort all shapes while keeping their area.

A caveat on its size and shape: the one equation det = 1 cuts dimension by exactly one, so dim SL(n,R) = n^2 - 1. Unlike SO(n) and U(n), SL(n,R) is NOT compact — shears like [1, t; 0, 1] have determinant 1 for every t, so you can run off to infinity inside the group. That non-compactness is why averaging over SL(n) needs more care than over O(n).