# Lazard correspondence establishes a correspondence between characteristic Lazard Lie subgroups and characteristic Lazard Lie subrings

From Groupprops

## Statement

Suppose is a Lazard Lie group, is its Lazard Lie ring, and and are the exponential and logarithm maps respectively (they are both bijections and are inverses of each other). Note that we may wish to think of and as having the same underlying set and treat the bijections as being the identity map on the underlying set; however, for conceptual convenience, we are using separate symbols for the group and Lie ring and explicit names for the bijections.

This bijection establishes a correspondence:

characteristic Lazard Lie subgroups of characteristic Lazard Lie subrings of