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Jul 2, 2015 at 6:23 vote accept Glasby
Jul 1, 2015 at 16:06 history edited Alireza Abdollahi CC BY-SA 3.0
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Jul 1, 2015 at 15:35 comment added Glasby Sorry Alireza. I think you wanted $G(d,p)$ to be $(d+1)\times(d+1)$ upper triangular matrices over ${\mathbb F}_p$ (not $d\times d$). Then $G(d,p)$ is a witness: $d$-generated, has exponent $p$ if $p> d$, and class $d$. Thus $c(d,p)\geq d$ for $p>d$. This answers Questions 1 and 2.
Jul 1, 2015 at 14:36 comment added Glasby Thanks Alireza. A lower bound can be established with a `witness': if $G$ is $d$-generated, has exponent $p$, and class $c$, then $c(d,p)\geq c$. For example $c(2,p)\geq 2$ if $p>2$; take $G$ to be extraspecial of exponent $p$. Your group $G(d,p)$ does not have exponent $p$ if $d>3$, and its class is $d-1$ not $d$. For Question 2 it suffices to find witnesses $G_{d,p}$ for $d=3,4$ and all $p\geq5$.
Jul 1, 2015 at 13:12 history answered Alireza Abdollahi CC BY-SA 3.0