= Lie group homomorphism determined by its differential
{c}
{title2=$\phi(\exp X)=\exp(d_e\phi(X))$}
A <Lie group homomorphism> intertwines <one-parameter subgroups> and therefore obeys the displayed identity. A connected <Lie group> is generated by an identity neighbourhood in an exponential chart, so its <homomorphism> is uniquely determined by the differential at the identity. On a disconnected <group>, only the restriction to its <identity component> is determined this way; the images of the other components require extra data.
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