Lazard correspondence establishes a 1-isomorphism between Lazard Lie group and Lazard Lie ring

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Statement

Suppose G is a Lazard Lie group and L is its Lazard Lie ring with \log:G \to L the logarithm map and \exp:L \to G the exponential map. (These are both bijections, and are inverses of each other).

Then, \log and \exp are 1-isomorphisms, i.e., they are isomorphisms when restricted to cyclic subgroups.

Facts used

  1. Logarithm map from Lazard Lie group to its Lazard Lie ring is a quasihomomorphism

Proof

Proof outline

This follows directly from fact (1), and the fact that the logarithm map is bijective.