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

powering-invariant characteristic subgroups of powering-invariant characteristic subrings of

## Related facts

- Lazard correspondence establishes a correspondence between characteristic Lazard Lie subgroups and characteristic Lazard Lie subrings
- Lazard correspondence establishes a correspondence between Lazard Lie subgroups and Lazard Lie subrings
- Lazard correspondence establishes a correspondence between abelian normal subgroups and abelian ideals
- Lazard correspondence establishes a correspondence between powering-invariant subgroups and powering-invariant subrings
- Lazard correspondence establishes a correspondence between powering-invariant normal subgroups and powering-invariant ideals