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

From Groupprops

## Contents

## Statement

Suppose is a Lazard Lie group and is its Lazard Lie ring with the logarithm map and the exponential map. (These are both bijections, and are inverses of each other).

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

## Facts used

## Proof

### Proof outline

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