2
$\begingroup$

I have received helpful answers to my two previous questions that focused on the symmetric group of degree 3 and the dihedral group of order 8. If $d$ is the minimal number of generators of a finite group, and $r$ is the minimal number of relations of the group, then in both examples $d = 2$, but for the given symmetric group we have $r = 2$ and for the given dihedral group we have $r = 3$. This holds for some generalizations of these groups as well.

If we focus now on 2-groups, I would like to know if there are any examples of finite 2-groups with $d = 2$ and $r = 2$, $d = 3$ and $r = 3$, or $d = 4$ and $r = 5$. This all goes back to sharpening the Golod/Shafarevich inequality, which says that for finite $p$-groups, $r > \large\frac{d^2}{4}$. I would like to know if in some particular cases (in particular if $d = 4$) if we can sharpen this further as $r > {\large\frac{d^2}{4}} + 1$ when $p = 2$.

$\endgroup$
0
$\begingroup$

Since asking my question I have learned that an example of a finite 2-group with d = 2 and r = 2 is the Quaternion group of order 8, with the presentation {a, b: a squared = b squared, (a inverse)ba = b inverse. I have also learned that it is known that there are finite 2-groups with d = 3 and r = 3 of orders 512, 1024 and 2048 (resp. 2 to the 9th, 10th, and 11th powers), as described in papers by authors including Fouladi, Jamali, Orfi, Havas, Newman, and O'Brien in 2004, 2008 and 2012 papers. Finally I have learned that a finite 2-group with d = 4 and r = 5 of order 16382 (2 the 14th power) was exhibited in a 2004 paper by Havas, Newman, and O'Brian, which was preceded by Havas and Newman in 1983 demonstrating four groups of the above type with orders 2 to the 16th, 17th, 18th, and 19th powers. Thus I have now answered all the questions I initially asked, although I don't like the answer that there exists a finite 2-group with d = 4 and r = 5, because it means that one cannot solve the infinite 2-class field tower conjecture for imaginary quadratic number fields with 2-class group of rank 4 by trying to sharpen the sharpened Golod/Shafarevich inequality. Oh well--such is math!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.