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

From Groupprops

## Statement

### For a group

Suppose is a group and are abelian normal subgroups of . Suppose is the join of subgroups . (Since they are all normal, this is equivalent to the product of subgroups ).

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

### For a subgroup

Suppose is a group and are abelian normal subgroups of . Let be the subgroup of defined as the join of subgroups . (Since they are all normal, this is equivalent to the product of subgroups ).

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

### Case of two subgroups

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