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?
2 Answers
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G5 has a presentation on generators a,b and relations ba=aab, abbabb=1.
answered Sep 9, 2010 at 2:20
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Martin Kassabov informed me that https://arxiv.org/abs/0804.1396 contains the fact for all $p$ (see Example 3.5 (2) on page 7). The presentation was discovered by B. Neumann in 1956.
