Group generated by finitely many abelian normal subgroups is nilpotent of class at most equal to the number of subgroups

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Statement

For a group

Suppose G is a group and H_1,H_2,\dots,H_n are abelian normal subgroups of G. Suppose G is the join of subgroups \langle H_1, H_2, \dots, H_n \rangle. (Since they are all normal, this is equivalent to the product of subgroups H_1H_2 \dots H_n).

Then, G is a nilpotent group of nilpotency class at most n.

For a subgroup

Suppose G is a group and H_1,H_2,\dots,H_n are abelian normal subgroups of G. Let H be the subgroup of G defined as the join of subgroups \langle H_1, H_2, \dots, H_n \rangle. (Since they are all normal, this is equivalent to the product of subgroups H_1H_2 \dots H_n).

Then, H is a nilpotent group of nilpotency class at most n. In particular, it is a nilpotent normal subgroup of G.

Case of two subgroups

In the case that n = 2, we get a group of nilpotency class two (in the group formulation) or a class two normal subgroup (in the subgroup formulation).