# Alternating bihomomorphism of finitely generated abelian groups arises as skew of 2-cocycle

## Contents

## Statement

### In terms of 2-cocycles

Suppose and are finitely generated abelian groups. Suppose is an alternating bihomomorphism of groups from to . Then, there exists a 2-cocycle for trivial group action such that , i.e.,:

### In terms of cohomology groups

Suppose and are finitely generated abelian groups. Consider the homomorphism:

which sends a cohomology class to the skew of any 2-cocycle representing it (that this is a homomorphism arises from the fact that skew of 2-cocycle for trivial group action of abelian group is alternating bihomomorphism). Then, this homomorphism is *surjective*, i.e., every alternating bihomomorphism arises from some cohomology class.

## Facts used

- Structure theorem for finitely generated abelian groups
- Symplectic decomposition of an alternating bilinear form taking values in a local principal ideal ring
- Orthogonal direct sum of cocycles is cocycle
- Symplectic decomposition of an alternating bilinear form taking values in integers

## Proof

### First part:reduction to the case where is either infinite cyclic or cyclic of prime power order

We first show that the problem can be reduced to the case that is a cyclic group.

**Given**: Finitely generated abelian groups and , an alternating bihomomorphism .

**To prove**: Assuming that we can solve the problem if were replaced by an infinite cyclic group or a cyclic group of prime power order, we can solve the problem in the general case, i.e., we can find a 2-cocycle such that .

**Proof**:

Step no. | Assertion/construction | Facts used | Given data/assumptions used | Previous steps used | Explanation |
---|---|---|---|---|---|

CR 1 | We can find cyclic groups such that , such that each is either infinite cyclic or cyclic of prime power order | Fact (1) | is finitely generated and abelian | Given+Fact direct | |

CR 2 | Let be the projection of on to the factor . Then, define . Each is an alternating bihomomorphism from to . | Step (CR1) | |||

CR3 | We can find cocycles such that . | Assumption that we can solve the reduced problem. | Step (CR1): Each is cyclic; Step (CR2): is an alternating bihomomorphism | ||

CR4 | Consider the mapping . is a 2-cocycle and . | Step (CR3) |

### Solution in the case of cyclic of prime power order

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