$A$ a subset of a finite group $G$ with strictly more than $|G|/2$ elements. Show $AA=G$.

Question

The question asks

(a) Let $A$ be a subset of finite group $G$ with strictly greater than $|G|/2$ elements. Show $AA=G$
and
(b) Show this can fail in a monoid.

I've been working on this for awhile but am not sure what to do...

Answer 1

(a) Let $g\in G$ and $A^{-1}:=\{a^{-1}|a\in A\}$. Then we know that $|gA^{-1}|=|A^{-1}|=|A|$, so $|gA^{-1}|+|A|>|G|$. Thus, $gA^{-1}$ and $A$ have a common element, and...

(b) Consider $(\mathbb{Z}/6\mathbb{Z}, \times)$. Let $A=\{0, 2, 3, 4\}$.