# Center is quotient-torsion-freeness-closed in nilpotent group

From Groupprops

## Statement

Suppose is a nilpotent group and is a prime number such that is -torsion-free. Suppose is the center of . Then the quotient group , which is also isomorphic to the inner automorphism group of , is also -torsion-free.

In other words, the center of is a quotient-torsion-freeness-closed subgroup of .

## Related facts

### Dual fact

For more on the background, see subgroup-quotient duality for groups.

The dual fact to this is derived subgroup is divisibility-closed in nilpotent group.

## Facts used

## Proof

The proof follows directly from Fact (1), specifically the (2) implies (6) implication, setting where is the nilpotency class of .