# Class two implies generated by abelian normal subgroups

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group of nilpotency class two) must also satisfy the second group property (i.e., group generated by abelian normal subgroups)

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## Statement

Suppose is a group of nilpotency class two. Then, is a group generated by abelian normal subgroups. In fact, is a *union* of abelian normal subgroups.

## Facts used

- Every group is a union of cyclic subgroups
- Abelian implies every subgroup is normal
- Cyclic over central implies abelian
- Normality satisfies inverse image condition

## Proof

**Given**: A group with center such that is an abelian group.

**To prove**: is a union of abelian normal subgroups of itself.

**Proof**:

Step no. | Assertion | Given data used | Facts used | Proof steps used | Explanation |
---|---|---|---|---|---|

1 | is a union of cyclic subgroups | -- | fact (1) | -- | |

2 | Each is normal in | is abelian | fact (2) | step (1) | |

3 | Let be the inverse image (for the quotient map with kernel ) in of . Then, each is abelian. | is the center of | fact (3) | step (1) | [SHOW MORE] |

4 | Each is normal in | -- | fact (4) | step (3) (construction of s) | |

5 | is the union of the s | -- | -- | steps (1) and (3) | [SHOW MORE] |