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.