On five well-known commutator identities

Abstract

We conjecture that five well-known identities universally satisfied by commutators in a group generate all such universal commutator identities. We use homological techniques to partially prove the conjecture.

Homology of multiplicative Lie rings

“…We recall the notion of multiplicative Lie ring due to Ellis [2], give some important examples and establish some structural results.…”

“…In [2] Ellis introduced multiplicative Lie algebras, which are called multiplicative Lie rings in the body of the current paper, to investigate an interesting combinatorial problem on group commutators. In [10] Point and Wantiez studied algebraic structural properties of multiplicative Lie algebras.…”

On the nilpotent multipliers of a group

“…It is explained (modulo notation) in [15] how the Magnus-Witt isomorphism leads to Proposition 2.10 [15] For any abelian group A there is a Lie ring isomorphism…”

Multiplicative Lie algebras

“…Thus, [x,y] [1]) Let E be a group and let P = E Z(E) . Define an action of P on E by for u ∈ E, x ∈ P (letting x = xZ(E), x ∈ E), x u = xux −1 .…”