I am aware of the theorem |G/Z(G)|=p with p prime implies G/Z(G) is cyclic and thus G is abelian, but I do not understand why. Is there not a theorem that says G abelian [itex]\Leftrightarrow[/itex] Z(G)=G? So what if |G|=p[itex]^{3}[/itex] and |Z(G)|=p[itex]^{2}[/itex]? This implies |G/Z(G)|=p implying G is abelian however G[itex]\neq[/itex]Z(G). What is the ambiguity here?(adsbygoogle = window.adsbygoogle || []).push({});

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# Group center properties

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