Let $p>2$ be a prime, $C\_p$ be the additive group of integers mod $p$. Then the multiplicative group $\{1,...,p-1\}$ of units in the field $Z/pZ$ is cyclic of order $p-1$, it acts on $C\_p$ by left multiplication. Let $G\_p$ be the corresponding semi-direct product of order $p(p-1)$. Question: does this group admit a presentation with 2 defining relations for some (all) $p > 3$? For $p=3$, $G\_p$ is the symmetric group $S\_3$ and it admits the balanced presentation $\langle a,b \mid a^2=1, aba=b^2\rangle$. (The question is attributed to Alex Lubotzky.) For example, what if $p=5$? Does $G\_5$, a cyclic-by-cyclic group of order 20, admit a presentation with 2 generators and 2 defining relations?



a balanced presentation of a cyclic-by-cyclic group?
G5 has a presentation on generators a,b and relations ba=aab, abbabb=1.


