Lazard correspondence up to isoclinism

Definition

The Lazard correspondence up to isoclinism is a slight variation up to isoclinism of the Lazard correspondence. It is weaker in the sense that it establishes an equivalence between groups up to isoclinism and Lie rings up to isoclinism. On the other hand, it is more general because there is a greater scope of permissible Lie rings and permissible groups that can be used.

In the case that the Lie ring is a Lazard Lie ring, the group on the other side is a Lazard Lie group, and vice versa. Further, the Lazard correspondence up to isoclinism in this case simply groups together multiple Lazard correspondences.

The permissible Lie rings are Lie rings for which both the derived subring and the quotient by the center are Lazard Lie rings. Further, the permissible groups are groups for which both the derived subgroup and the quotient by the center are Lazard Lie groups.

The correspondence is as follows: A Lie ring $L$ is identified with a group $G$ via a pair of isomorphisms:

• An isomorphism $\zeta$ of Lie rings between the Lie ring $L/Z(L)$ and the Lazard Lie ring of the inner automorphism group $G/Z(G)$ (which as a set we identify with $G/Z(G)$), and
• An isomorphism $\phi$ of Lie rings between the Lie ring $[L,L]$ and the Lazard Lie ring of the derived subgroup $[G,G]$

such that for $x,y\in L$, with images ${\overline {x}},{\overline {y}}$ mod $Z(L)$, we have:

$\phi (F(x,y))=[\zeta ({\overline {x}}),\zeta ({\overline {y}})]$

where $F$ on the left stands for the formula for group commutator in terms of Lie bracket and the bracket on the right is the group commutator, well defined because the group commutator in $G$ of two elements depends only on their cosets mod $Z(G)$.

The fact that this operation does not involve any division by forbidden primes follows indirectly from bound on prime power divisors of denominators in formula for group commutator in terms of Lie bracket.

In the p-group case

For p-groups (and in particular for finite p-groups), the Lazard correspondence up to isoclinism is a correspondence:

$p$-Lie rings of (3-local) class at most $p$ up to isoclinism $\leftrightarrow$ $p$-groups of (3-local) class at most $p$ up to isoclinism