21.3 (2026)

Let $G = A_n$ or $S_n$ and let $H, K$ be soluble subgroups of $G$. For all sufficiently large $n$, can we always find an element $x \in G$ such that $H \cap K^x = 1$? Does this hold for all $n \geqslant 21$?

Note that the conclusion is false when $G = S_{20}$ and $H = K = (S_4 \wr S_4) \times S_4$.