Let $a,b$ be two positive integer numbers. A group $G$ is called *$a{\times}b$-decomposable* if there are subsets $A,B\subset G$ of cardinality $|A|=a$ and $|B|=b$ such that $AB=G$ where $AB=\{xy:x\in A,\;y\in B\}$.

I am looking for an example of a finite group $G$ which is not $a{\times}b$-decomposable for some numbers $a,b$ with $a\cdot b=|G|$.

**Remark.** It is easy to see that a group $G$ is $a{\times}b$-decomposable if $a\cdot b=|G|$ and $G$ contains a subgroup of order or index equal to $a$ or $b$. Consequently, any Abelian group $G$ is $a{\times}b$-decomposable for any numbers $a,b$ with $a\cdot b=|G|$.

According to the answer of Geoff Robinson to this MO-problem the alternating group $A_9$ contains no subgroups of order or index equal to 35.

Question 1.Is the group $A_9$ $35{\times}5184$-decomposable?

By the comments of @YCor to the same MO-problem,

$\bullet$ the group $PSL_2(11)$ has cardinality $|PSL_2(11)|=15\times 44$ but contains no subgroups of order or index 15;

$\bullet$ the group $PSL_2(13)$ has cardinality $|PSL_2(13)|=21\times 52$ but contains no subgroups of order or index 21.

Question 2.Is the group $PSL_2(11)$ $15{\times}44$-decomposable?

Question 3.Is the group $PSL_2(13)$ $21{\times}52$-decomposable?

subsets, not necessarily subgroups.) $\endgroup$ – bof Nov 25 '18 at 2:28