# 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.

## Contents

- 1 Statement
- 2 Related facts
- 3 Proof (cohomology interpretation)
- 4 Proof (hands-on)
- 4.1 Comment on the operation being well-defined
- 4.2 Addition is associative
- 4.3 Addition is commutative
- 4.4 Agreement of identity and inverses
- 4.5 The Lie bracket is additive in the first variable
- 4.6 The Lie bracket is additive in the second variable
- 4.7 The Lie bracket is alternating
- 4.8 The Jacobi identity and class two

## Statement

Suppose is a class two group. Suppose is a central subgroup of and is an abelian group (in particular, this means that . is a function such that is constant in each input on the cosets of . Denote by the induced function . Suppose the following four conditions are satisfied:

- The
**cocycle condition**: This states that for all . This is equivalent to requiring that be a 2-cocycle from to . - The
**skew equals commutator condition**: This states that . - The
**identity-preservation condition**: This states that for all , where is the identity element. - The
**inverse-preservation condition**: This states that for all , where is the identity element.

We give the structure of a Lie ring as follows:

- The addition is given by:

- 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, acquires the structure of a class two Lie ring.

### Additional claims

- We say that is
**cyclicity-preserving**if, whenever satisfy the property that is cyclic, then . If is cyclicity-preserving, the identification between and the additive group of its generalized Baer Lie ring for gives a 1-isomorphism of groups. In particular, is 1-isomorphic to an abelian group.

### Note

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

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## Related facts

- Proof of generalized Baer construction for group for class two Lie ring with a suitable cocycle
- Uniqueness theorem for generalized Baer construction using cyclicity-preserving 2-cocycle

## Proof (cohomology interpretation)

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## Proof (hands-on)

### Comment on the operation being well-defined

Since 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**:

**Key proof ingredient**: The cocycle condition.

**Proof**: [SHOW MORE]

### Addition is commutative

**To prove**:

**Key proof ingredient**: Skew is commutator.

**Proof**: [SHOW MORE]

### Agreement of identity and inverses

**To prove**: If is the identity element for the multiplication, then and .

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

**Proof**: [SHOW MORE]

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

**To prove**:

**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 is trivial for all , the Lie bracket also satisfies the same condition. Thus, it satisfies Jacobi's identity and also the condition for class two.