What is an example of a finite centerless group with at least 3 generators? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:25:54Z http://mathoverflow.net/feeds/question/53537 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/53537/what-is-an-example-of-a-finite-centerless-group-with-at-least-3-generators What is an example of a finite centerless group with at least 3 generators? Makhalan Duff 2011-01-27T21:07:38Z 2011-02-07T18:40:26Z <p>I need one to test a theory. There are probably many, but I can't seem to think of a single one. My guess is examples are pretty big.</p> <p>Is there a systematic way to find such examples? Are there databases one can go through to find these things?</p> http://mathoverflow.net/questions/53537/what-is-an-example-of-a-finite-centerless-group-with-at-least-3-generators/53540#53540 Answer by Mark Sapir for What is an example of a finite centerless group with at least 3 generators? Mark Sapir 2011-01-27T21:25:01Z 2011-01-28T00:34:42Z <p>The group $S_3\wr ({\mathbb Z}_2 \times {\mathbb Z}_2 \times {\mathbb Z}_2)$ where $S_3$ is the symmetric group with 6 elements, ${\mathbb Z}_2$ is the group with 2 elements, $\wr$ is the wreath product. </p> <p>The fact that it does not have a center is proved by inspection. The fact that it needs at least 3 generators follows from the fact that ${\mathbb Z}_2^3$ is its quotient. There are lots of similar examples of course. </p> <p><b> Update. </b> In general , if you take any centerless group $G$ and any group $H$ that needs at least 3 generators, then the wreath product $G\wr H$ has both properties (is centerless and needs at least 3 generators). Another way to construct examples is (as Derek Holt comment below shows) to take any centerless finite group $G$ with nontrivial abelianization and take $G\times G\times G$. </p> http://mathoverflow.net/questions/53537/what-is-an-example-of-a-finite-centerless-group-with-at-least-3-generators/53554#53554 Answer by Derek Holt for What is an example of a finite centerless group with at least 3 generators? Derek Holt 2011-01-27T23:30:45Z 2011-01-27T23:30:45Z <p>The smallest example has order 18. It has a normal elementary abelian subgroup $N$ of order 9, and an element $t$ of order 2 such that $txt=x^{-1}$ for all $x \in N$.</p> http://mathoverflow.net/questions/53537/what-is-an-example-of-a-finite-centerless-group-with-at-least-3-generators/53577#53577 Answer by Zsbán Ambrus for What is an example of a finite centerless group with at least 3 generators? Zsbán Ambrus 2011-01-28T08:11:42Z 2011-02-07T18:40:26Z <p>I can't prove this, but the Rubik's cube group might work.</p> <p>Update: so this doesn't work. Thanks for the comments.</p>