(F.g., f.p.) groups with exactly $n$ normal subgroups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T04:23:30Z http://mathoverflow.net/feeds/question/73571 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73571/f-g-f-p-groups-with-exactly-n-normal-subgroups (F.g., f.p.) groups with exactly $n$ normal subgroups Victor 2011-08-24T15:00:46Z 2011-08-24T18:25:57Z <p>Dear All!</p> <p>I tried for several evenings to find an answer to the following basic question and I cannot see what is the answer:</p> <p>Given an integer $n\geq 3$, does there exist an (infinite) group with exactly $n$ normal subgroups?</p> <p>If "yes", what about the same questions for finitely generated groups, finitely presented groups?</p> <p>I guess this must have been done.</p> http://mathoverflow.net/questions/73571/f-g-f-p-groups-with-exactly-n-normal-subgroups/73574#73574 Answer by Daniel Litt for (F.g., f.p.) groups with exactly $n$ normal subgroups Daniel Litt 2011-08-24T15:16:18Z 2011-08-24T15:16:18Z <p>In general, $\mathbb{Z}/2^k\mathbb{Z}$ has $k+1$ normal subgroups, namely $\mathbb{Z}/2^j\mathbb{Z}$ for $0\leq j\leq k$. So the answer to your question is "yes."</p> http://mathoverflow.net/questions/73571/f-g-f-p-groups-with-exactly-n-normal-subgroups/73580#73580 Answer by Mark Sapir for (F.g., f.p.) groups with exactly $n$ normal subgroups Mark Sapir 2011-08-24T15:50:25Z 2011-08-24T18:25:57Z <p>If $n$ is even, then the answer is "yes". Take the direct product of a simple (infinite) group and ${\mathbb Z}/2^j{\mathbb Z}$. Every normal subgroup either is inside the finite cyclic group or contains the simple group. Total number is twice the number of normal subgroups of the cyclic group.If $n$ is odd, you would need to take a cyclic central extension of a simple group. That is also possible (the simple group can be, say, the Tarski monster, see our paper with Olshanskii and Osin on Lacunary hyperbolic groups in the arXiv).</p> <p><b> Edit. </b> If you want f.p. groups, look at the central extensions of the Thompson group $T$ described <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.80.628&amp;rep=rep1&amp;type=pdf" rel="nofollow">here</a>.</p>