What is an example of a finite centerless group with at least 3 generators? - MathOverflow

What is an example of a finite centerless group with at least 3 generators?

2011-01-27T21:07:38Z

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.

Is there a systematic way to find such examples? Are there databases one can go through to find these things?

Answer by Mark Sapir for What is an example of a finite centerless group with at least 3 generators?

2011-01-27T21:25:01Z

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.

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.

Update. 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 $G\times G\times G$.

Answer by Derek Holt for What is an example of a finite centerless group with at least 3 generators?

2011-01-27T23:30:45Z

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 $x \in N$.

Answer by Zsbán Ambrus for What is an example of a finite centerless group with at least 3 generators?

2011-01-28T08:11:42Z

I can't prove this, but the Rubik's cube group might work.

Update: so this doesn't work. Thanks for the comments.