= Universal covering Lie group
{title2=$p:\widetilde G\longrightarrow G^\circ$}
The <universal cover> of a connected <Lie group> carries a unique lifted <Lie group> structure with a chosen identity, making its <covering map> a <Lie group homomorphism>. The kernel is a discrete central <subgroup>: conjugation of any kernel element is a continuous map from a connected <group> into a discrete set, hence constant. Every connected <Lie group> is therefore a quotient of its <simply connected> cover by a discrete central <subgroup>.
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