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

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Statement

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

Examples

Let p be an odd prime. Consider groups of order p^3:

Related facts

Converse

Other related facts


Facts used

  1. Logarithm map from Lazard Lie group to its Lazard Lie ring is a 1-isomorphism

Proof

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