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 Simple(st) 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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    $\begingroup$ en.wikipedia.org/wiki/Tarski_monster_group . I assume that by "no finite normal subgroups" you mean "no finite nontrivial normal subgroups". $\endgroup$ – Martin Brandenburg May 11 '12 at 11:34
  • $\begingroup$ Note that Tarski monsters referenced by Martin are also 2-generated and simple. $\endgroup$ – Misha May 11 '12 at 13:04

The group $G$ described in " Simple(st) 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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  • $\begingroup$ Thanks! What does that vertical squiggly line mean in the second example? $\endgroup$ – Dori Bejleri May 11 '12 at 19:15
  • $\begingroup$ it means "wreath product". $\endgroup$ – YCor May 12 '12 at 17:54

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