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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asked May 11, 2012 at 11:20
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5$\begingroup$ en.wikipedia.org/wiki/Tarski_monster_group . I assume that by "no finite normal subgroups" you mean "no finite nontrivial normal subgroups". $\endgroup$– Martin BrandenburgCommented May 11, 2012 at 11:34
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$\begingroup$ Note that Tarski monsters referenced by Martin are also 2-generated and simple. $\endgroup$– MishaCommented May 11, 2012 at 13:04
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1 Answer
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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? $\endgroup$ Commented May 11, 2012 at 19:15
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