A finite $p$-group $G$ always has a non-trivial center. However, there exist infinite $p$-groups with trivial center, see for example this question http://mathoverflow.net/questions/49349/simplest-example-of-an-infinite-p-group-with-trivial-center. My question is do there exist such infinite $p$-groups with trivial center that contain no finite normal subgroups? More generally, does there exist an infinite $p$-group $G$ such that infinitely many elements of $G$ are not in $Z(G)$ and such that $G$ has no finite normal subgroups?
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The group $G$ described in " http://mathoverflow.net/questions/49349/simplest-example-of-an-infinite-p-group-with-trivial-center " is in fact an ICC group. This means that every non-trivial element $g\in G$ has an infinite conjugacy class. As a consequence, $G$ has trivial center (because the conjugacy class of an element in the center contains only one element), and no finite normal subgroups (because every normal subgroup is a union of conjugacy classes). The group $\mathbb{Z}/p\wr(\mathbb{Z}/p)^\infty$ is also an ICC (infinite conjugacy classes) p-group, so its center is trivial and it can not have finite normal subgroups. |
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