Every Two Element in A Coset

Question

For every $a,b \in G$, $a,b \in cH$ for some maximal subgroup $H$ of $G$ and some $c \in G$.

For what groups is the following property true? I know its true for $\mathbb{Z_m} \times \mathbb{Z_n}$ but what about $S_n$ or $A_n$? For $n = 3,4$, I have verified that this is true but is this true generally?

Answer 1

$a,b\in cH$ iff $aH=cH$ and $a^{-1}b\in H$.

So the proposition is true iff every element $g=a^{-1}b$ of $G$ is contained in a maximal subgroup of $G$. If $G$ is finite, that's true iff $G$ is not cyclic.

Answer 2

If $a=b$ the question is the same as asking "Do every group have a maximal subgroup?" And some groups (like $\mathbb{Q}$) do not have this property.