Cyclic and capable implies trivial

Statement
Suppose $$G$$ is a fact about::cyclic group that is also a fact about::capable group: there exists a group $$H$$ such that the fact about::inner automorphism group $$H/Z(H)$$ is isomorphic to $$G$$. Then, $$G$$ is trivial.

Facts used

 * 1) uses::Cyclic over central implies abelian

Proof
Given: A group $$H$$ such that $$H/Z(H)$$ is cyclic.

To prove: $$H/Z(H)$$ is trivial.

Proof: By fact (1), $$H$$ is abelian. Since any abelian group equals its own center, we obtain that $$Z(H) = H$$, so $$H/Z(H)$$ is trivial.