Class two implies generated by abelian normal subgroups

From Groupprops
Jump to: navigation, search
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)
View all group property implications | View all group property non-implications
Get more facts about group of nilpotency class two|Get more facts about group generated by abelian normal subgroups

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.

Facts used

  1. Every group is a union of cyclic subgroups
  2. Abelian implies every subgroup is normal
  3. Cyclic over central implies abelian
  4. Normality satisfies inverse image condition

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_is)
5 P is the union of the P_is -- -- steps (1) and (3) [SHOW MORE]