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

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

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

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

  1. Equivalence of definitions of nilpotent group that is torsion-free for a set of primes


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