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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

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    Ok, this time I think I understood: first, it isn't true that $\,G/Z(G)\,$ has to be abelian (for an easy counter example take $\,G=S_3\,$) , so what are your assumptions and what is your question?2012-10-28
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    my assumptions is exp(G/Z(G))=2 Donatonio not that exp(S_3)=62012-10-28
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    Ok. So you *assume* that $\,G/Z(G)\,$ is abelian of exponent 2? And then, of course, $\,G\,$ is nilpotent of class 2. What do you mean by $\,O(x)\,$?2012-10-28

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