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

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## Statement

Suppose $G$ is a Lazard Lie group, $L$ is its Lazard Lie ring, and $\exp:L \to G$ and $\log:G \to L$ 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 $G$ and $L$ 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 characteristic subgroups of $G$ $\leftrightarrow$ powering-invariant characteristic subrings of $L$