Unitary group ($U(n)$)

Group of the unitary matrices.

Complex analogue of the orthogonal group.

One notable difference from the orthogonal group however is that the unitary group is connected "because" its determinant is not fixed to two disconnected values 1/-1, but rather goes around in a continuous unit circle. $U(1)$ is the unit circle.

Diffeomorphic to the 3 sphere.

The $U(1)$ unitary group is one very over-generalized way of looking at it :-)

The complex analogue of the special orthogonal group, i.e. the subgroup of the unitary group with determinant equals exactly 1 instead of an arbitrary complex number with absolute value equal 1 as is the case for the unitary group.

Double cover of $SO(3)$.

Isomorphic to the quaternions.

Bibliography:

Pauli matrix.