Let $m\in \mathbb{N}$. Show that there exists a finite group $G$ with $|G|\gt m$ such that every subset $S$ of $G$ with $|S| \lt \log_2|G|$ is not a a generating set of $G$.
I find this question kinda wierd, as I can prove that there exists a subgroup $S$ with $|S| \le \log_2|G|$ which is a generating set of $G$ (by taking $<x_1>, <x_1,x_2> \dots$ until we get that generating set...)
So how could it be?
Let $m\in\mathbb{N}$. Let $G_n=(\mathbb{Z}/2)^m$. Then $|G|=2^m$, so $log_2|G|=m$. The $\mathbb{Z}$-module structure (i.e. the group structure) on G induces a $\mathbb{Z}/2$-module structure, which is nothing a vectorspace-structure. But this vectorspace has dimension m, so any set $S$ with $|S|<m$ can not generate the whole group $G$.