Proof of generalized Baer construction of Lie ring for class two 2-group with a suitable cocycle

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This proof is instrumental in the definitions of generalized Baer IIP Lie ring and generalized Baer cyclicity-preserving Lie ring.

Statement

Suppose P is a class two group. Suppose A is a central subgroup of P and G \cong A/P is an abelian group (in particular, this means that [P,P] \le A \le Z(P). f is a function f:P \times P \to A such that f is constant in each input on the cosets of A. Denote by \overline{f} the induced function P/A\times P/A \to A. Suppose the following four conditions are satisfied:

  1. The cocycle condition: This states that f(x,y)f(xy,z) = f(x,yz)f(y,z) for all x,y,z \in P. This is equivalent to requiring that \overline{f} be a 2-cocycle from P/A to A.
  2. The skew equals commutator condition: This states that f(x,y)(f(y,x))^{-1} = [x,y].
  3. The identity-preservation condition: This states that f(x,e) = f(e,x) = e for all x, where e is the identity element.
  4. The inverse-preservation condition: This states that f(x,x^{-1}) = e for all x, where e is the identity element.

We give P the structure of a Lie ring as follows:

  • The addition is given by:

x + y := \frac{xy}{f(x,y)}

  • The additive identity for the Lie ring is the group's identity element.
  • Additive inverses are the same as multiplicative inverses in the group.
  • The Lie bracket is the same as the commutator in the group.

The claim is that with these operations, P acquires the structure of a class two Lie ring.

Additional claims

  1. We say that f is cyclicity-preserving if, whenever x,y \in P satisfy the property that \langle x,y \rangle is cyclic, then f(x,y) = e. If f is cyclicity-preserving, the identification between P and the additive group of its generalized Baer Lie ring for f gives a 1-isomorphism of groups. In particular, P is 1-isomorphic to an abelian group.

Note

Note that the statement and construction work for odd-order p-groups as well, but in those cases, f is uniquely determined and is denoted as \sqrt{[x,y]}. This is the Baer correspondence. For full proof, refer: Proof of Baer construction of Lie ring for odd-order class two p-group

The proof can be given using an interpretation in terms of twisting the cohomology class of the extension. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Related facts

Proof (cohomology interpretation)

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Proof (hands-on)

Comment on the operation being well-defined

Since f(x,y) is central, we do not need to specify, when dividing by it, whether we are dividing on the left or the right. Thus, the fraction notation is unambiguous.

Addition is associative

To prove: \! (x + y) + z = x + (y + z)

Key proof ingredient: The cocycle condition.

Proof: [SHOW MORE]

Addition is commutative

To prove: \! x + y = y + x

Key proof ingredient: Skew is commutator.

Proof: [SHOW MORE]

Agreement of identity and inverses

To prove: If e is the identity element for the multiplication, then x + e = e + x = x and x + x^{-1} = x^{-1} + x = e.

Key proof ingredient: Identity-preservation and inverse-preservation.

Proof: [SHOW MORE]

The Lie bracket is additive in the first variable

To prove: \! [x + y,z] = [x,z] + [y,z]

Proof: [SHOW MORE]

The Lie bracket is additive in the second variable

This is analogous to additivity in the first variable, as shown above.

The Lie bracket is alternating

This follows from the fact that the commutator of an element with itself is the identity.

The Jacobi identity and class two

Since the Lie bracket coincides with the commutator, and the commutator satisfies that [[x,y],z] is trivial for all x,y,z, the Lie bracket also satisfies the same condition. Thus, it satisfies Jacobi's identity and also the condition for class two.