Normal closure of 2-subnormal subgroup of prime order in nilpotent group is abelian

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a fact about::2-subnormal subgroup of $$G$$ of order $$p$$ for some prime number $$p$$. Then, the fact about::normal closure of $$H$$ in $$G$$ is an abelian group.

Stronger facts

 * Weaker than::Normal closure of 2-subnormal subgroup of prime order is abelian: We can drop the assumption of the whole group being nilpotent.
 * Weaker than::Normal closure of 2-subnormal subgroup of prime power order in nilpotent group has nilpotency class at most equal to prime-base logarithm of order

Opposite facts

 * Normal closure of 3-subnormal subgroup of prime order in nilpotent group need not be abelian

Facts used

 * 1) uses::Minimal normal implies central in nilpotent