## Dense set in finite field [closed]

Hi everyone

My question seems classic, so it might be trivial or simple, therefore I am sorry in this case. Let $A\subseteq \mathbb{F}_p$ be a set with size $a$, and let $G$ be the subgroup generated by $A$. I am interested in $|G|$ for instance how big is it?. I guess if $|A|\gg \sqrt{p}$ then $G=\mathbb{F}_p$

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Do you mean $p$ to be prime? Then $G$ is either trivial or $\mathbb{F}_p$. – S. Carnahan Apr 9 2011 at 4:31
What if $p=17$, $|G|=|A|=8$ (I assume that the operation in $G$ is multiplication)? $8>\sqrt{17}$. – Mark Sapir Apr 9 2011 at 4:33
P is prime and operator is multiplication, otherwise question is trivial. – M.B Apr 9 2011 at 4:36
If $p=2^n+1$, then $\mathbb{F}_p^*$ has subgroups of orders $2^k$ for every $k$ between $0$ and $n$. – Mark Sapir Apr 9 2011 at 4:42