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Let $G$ be a group. Let $Z(G)$ be the center of $G$, the set of elements that commute with every element of $G$.

Then, can we say that there is some elements in $Z(G/Z(G))$ which is not $Z(G)$?

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    The sets $Z(G/Z(G))$ and $Z(G)$ are disjoint... they do not even share the identity element!2012-04-19
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    @Mariano: "which is not", not "which is not in". $Z(G)$ is the identity element of $G/Z(G)$.2012-04-19
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    Oooooooooh. ${}$2012-04-19
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    Recall that any finite $p$-group has nontrivial center, so let $G$ be a nonabelian $p$-group...2012-04-19

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