# Lazard correspondence establishes a correspondence between powering-invariant subgroups and powering-invariant subrings

From Groupprops

This article describes a bijection, i.e., a correspondence, that arises as part of the Lazard correspondence.

## 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:

powering-invariant subgroups of powering-invariant subrings of