From Section "Lie algebra of a isometry group" we reach:

Group of rotations of a rigid body.

Like orthogonal group but without reflections. So it is a "special case" of the orthogonal group.

This is a subgroup of both the orthogonal group and the special linear group.

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Based on the $L_x$,$L_y$ and $L_z$ derived at Lie algebra of $SO(3)$ we can calculate the Lie bracket as:

Has $SU(2)$ as a double cover.

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 :-)

Analysis of some of them follows.

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.