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

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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

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