Editing Covering group (section)

== Examples ==
* The universal covering group of the [[circle group]] '''T''' is the additive group of [[real number]]s ('''R''', +) with the covering homomorphism given by the mapping {{nowrap|'''R''' → '''T''' : ''x'' ↦ exp(2''πix'')}}. The kernel of this mapping is isomorphic to Z.
* For any integer ''n'' we have a covering group of the circle by itself {{nowrap|'''T''' → '''T'''}} that sends ''z'' to {{itco|''z''}}<sup>''n''</sup>. The kernel of this homomorphism is the [[cyclic group]] consisting of the ''n''th [[roots of unity]].
* The rotation group [[SO(3)]] has as a universal cover the group [[SU(2)]], which is isomorphic to the group of [[versor]]s in the quaternions. This is a double cover since the kernel has order 2. (cf the [[tangloids]].)
* The [[unitary group]] U(''n'') is covered by the compact group {{nowrap|'''T''' × SU(''n'')}} with the covering homomorphism given by {{nowrap|1=''p''(''z'', ''A'') = ''zA''}}. The universal cover is  {{nowrap|'''R''' × SU(''n'')}}.
* The [[special orthogonal group]] SO(''n'') has a double cover called the [[spin group]] Spin(''n''). For {{nowrap|''n'' ≥ 3}}, the spin group is the universal cover of SO(''n'').
* For {{nowrap|''n'' ≥ 2}}, the universal cover of the [[special linear group]] {{nowrap|SL(''n'', '''R''')}} is ''not'' a [[matrix group]] (i.e. it has no faithful finite-dimensional [[group representation|representation]]s).