Class two implies generated by abelian normal subgroups

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

Proof

Given: A group $P$ with center $C$ such that $G := P/C$ is an abelian group.

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

Proof:

Step no. Assertion Given data used Facts used Proof steps used Explanation
1 $G$ is a union of cyclic subgroups $G_i, i \in I$ -- fact (1) --
2 Each $G_i$ is normal in $G$ $G$ is abelian fact (2) step (1)
3 Let $P_i$ be the inverse image (for the quotient map $P \to G$ with kernel $C$) in $P$ of $G_i$. Then, each $P_i$ is abelian. $C$ is the center of $P$ fact (3) step (1) [SHOW MORE]
4 Each $P_i$ is normal in $P$ -- fact (4) step (3) (construction of $P_i$s)
5 $P$ is the union of the $P_i$s -- -- steps (1) and (3) [SHOW MORE]