Can you show that $\,exp(G/Z(G))=2 \Longrightarrow exp(G')=2$ ?
My try is: clearly $\,G/Z(G)\,$ is abelian group,so G is nilpotent of class 2 and we have [G',G]=1 if x ϵ G' then [x,g]=1 ∀ g ϵ G so G'≤Z(G).
But I can't show that O(x)=2
Can you show that $\,exp(G/Z(G))=2 \Longrightarrow exp(G')=2$ ?
My try is: clearly $\,G/Z(G)\,$ is abelian group,so G is nilpotent of class 2 and we have [G',G]=1 if x ϵ G' then [x,g]=1 ∀ g ϵ G so G'≤Z(G).
But I can't show that O(x)=2