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.