A property for some finite groups (especially ${\rm PSL}(2,13)$)

Question

Let $G$ be a finite group of order $n$ and consider the following property:

(P) for every factorization $n=ab$ there exist subsets $A$ and $B$ such that $|A|=a$, $|B|=b$ and $G=AB$.

($AB=\{ab:a\in A, b\in B\}$)

Note. If $G$ has the property that for every divisor $d$ of $n$ there exists a subgroup of $G$ with order $d$ or $n/d$, then we can show that $G$ enjoys (P).

Therefore, (P) is true for all finite abelian groups, and also one can check that the groups $S_n$, $A_n$ and ${\rm PSL}(2,n)$, where $n\leq 8$, have the property.

Now, is (P) true for ${\rm PSL}(2,13)$?

(we think this is a good candidate for a probable counterexample.)

Thanks in advance.

Answer 1

I don't know whether this can be generalized in any way, but for the $G=PSL_2(13)$ case, you can get sets $A$ and $B$ of sizes $21$ and $52$ with the property $G=AB$ by simply multiplying Sylow-$3$ and Sylow-$7$ subgroups for $A$ and Sylow-$2$ and Sylow-$13$ subgroups for $B$, as GAP shows:

gap> G:=PSL(2,13);;
gap> A := Concatenation(List(SylowSubgroup(G, 7), s -> AsList(RightCoset(SylowSubgroup(G, 3), s))));;
gap> B := Concatenation(List(SylowSubgroup(G, 13), s -> AsList(RightCoset(SylowSubgroup(G, 2), s))));;
gap> Size(G);
1092
gap> Size(A);
21
gap> Size(B);
52
gap> Size(Concatenation(List(A, a -> List(B, b -> a*b))));
1092