# Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring

From Groupprops

## Contents

## Statement

Suppose is a Lazard Lie group and is the Lazard Lie ring for , i.e., the Lie ring obtained by applying Lazard's theorem. Then, the additive group of has the same order statistics as . In other words, and the additive group of are Order statistics-equivalent finite groups (?).

## Examples

Let be an odd prime. Consider groups of order :

- For the non-abelian group of exponent p, the additive group of the Lazard Lie ring is the elementary abelian group of order . Thus, both these groups have the same order statistics.
- For the non-abelian group of prime-squared exponent, the additive group of the Lazard Lie ring is the direct product of the cyclic group of order and the cyclic group of order .

## Related facts

### Converse

- Order statistics of a finite group determine whether it is nilpotent
- Finite abelian groups with the same order statistics are isomorphic
- Finite group having the same order statistics as a cyclic group is cyclic

## Facts used

## Proof

This follows directly from fact (1), i.e., the fact that the bijective logarithm map from to restrict to isomorphisms on cyclic subgroups.