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

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.

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]

## 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.

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

Key proof ingredient: The cocycle condition.

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

Key proof ingredient: Skew is commutator.

### 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.

### The Lie bracket is additive in the first variable

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

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.