unitary group U(n), SU(n)
The unitary group is the complex cousin of the orthogonal group. Where O(n) preserves real lengths, U(n) preserves the complex (Hermitian) inner product on C^n: it is the group of complex rotations. It is the symmetry group quantum mechanics is built on, because unitary maps preserve probabilities.
Formally A is unitary iff A^* A = I, where A^* = conj(A)^T is the conjugate transpose. Equivalently the columns are an orthonormal basis of C^n under <z,w> = sum z_k conj(w_k). Then |det A| = 1, so det A lies on the unit circle. The special unitary group SU(n) demands det A = +1 exactly — the determinant phase is pinned down.
Geometrically U(n) = { maps preserving <Az, Aw> = <z, w> over C }. Unlike O(n), U(n) is CONNECTED: the determinant can be any phase e^{i theta}, and you can rotate that phase continuously to 1, so there is no two-piece split. U(n) and SU(n) are both compact, which again makes averaging (Haar measure) well-behaved.
Why physicists care: SU(2) governs spin, SU(3) the strong interaction, U(1) electromagnetism's phase. A caveat to keep straight: U(n) sits inside GL(n,C) as a real manifold of dimension n^2, while SU(n) has dimension n^2 - 1 (the single det = 1 constraint removes one real direction). These are real dimensions, not complex.
A general SU(2) element is built from two complex numbers a, b with |a|^2 + |b|^2 = 1 — exactly a point on the 3-sphere.
Sanity check: U(1) = { e^{i theta} } is just the unit circle, the simplest nontrivial compact group. SU(2) turns out to be the 3-sphere S^3, sitting one step above SO(3) as its double cover — that is the bridge to quaternions.