3
$\begingroup$

Is there a presentation with two generators and two relators for the group $C_4 \cdot D_8$? This group is of order 32 and its IdSmallGroup in GAP is [32,15]. Also it has the following presentation with 2 generators and 3 relators:

$\langle x,y \;|\; y^x=y^3, (y^2)^x=y^{-2}, yx^{-1}y=x^3\rangle$

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

$$\langle x,y \mid y^x=y^3, yx^{-1}y=x^{-5}\rangle.$$

I found that presentation by trial and error, but minimal presentations are known for $2$-groups of order up to $64$. See

Sag, T. W.; Wamsley, J. W. Minimal presentations for groups of order $2^n$, $n \le 6$, J. Austral. Math. Soc. 15 (1973), 461–469.

Improve this answer
$\endgroup$
9
  • $\begingroup$ @ Many thanks. Could you please give a bit explanation how you get this presentation? $\endgroup$
    – Alireza Abdollahi
    Commented Jun 18, 2015 at 8:12
  • $\begingroup$ In the reference, $2n$ should be $2^n$. $\endgroup$
    – Alireza Abdollahi
    Commented Jun 18, 2015 at 10:11
  • 2
    $\begingroup$ @AlirezaAbdollahi Often such presentations are found by brute force search : For a generating set, run through all words up to a certain length to see which are relators, then take combinations of relators and see whether a coset enumeration happens to terminate and prove the result. $\endgroup$
    – ahulpke
    Commented Jun 18, 2015 at 15:08
  • 1
    $\begingroup$ My search didn't really take long. I tried missing out the second relator and found that gave a group of order $96$, which looked promising. Since $x$ and $y$ should both have order $8$, I tried replacing the $x^3$ in the original presentation by $x^{-5}$, knowing that tricks like that often work. $\endgroup$
    – Derek Holt
    Commented Jun 18, 2015 at 17:55
  • 1
    $\begingroup$ I am not aware of any work on minimal presentations of $2$-groups that goes further than $2^6$, which was done by Wamsley a long time ago. BTW, the groups of order $2^{10}$ are not currently in the GAP library. The total number of such groups is known, but they have not been listed individually. $\endgroup$
    – Derek Holt
    Commented Jun 19, 2015 at 7:31

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .