Let H be a subgroup of Sym(n) that is not too large - say $|H| = |Sym(n)|^{o(1)}$. (A group that is especially interesting in this context is the abelian group H = {products of powers of $\pi_1$,...,$\pi_k$}, where $\pi_1$,$\pi_2$,...,$\pi_k$ are disjoint cycles of distinct lengths, k about $\sqrt{n}$, each length also about $\sqrt{n}$.)

If I conjugate H by a few random elements, will H stick in different directions? To make this precise: will the map

$H \times H \times ... \times H \to Sym(n)$ given by

$(h_1,h_2,...,h_l)\mapsto g_1 h_1 g_1^{-1} ... g_l h_l g_l^{-1}$ ($l=O(1)$)

be in some sense close to being injective for g_1,...,g_l random elements of Sym(n)? (By "close to being injective", I mean, say, that if two tuples $(h_1,h_2,...,h_l)$, $(h_1',h_2',...,h_l')$ map to the same element of Sym(n), then one of the elements $h_1 h_1'^{-1}$, ... , $h_l h_l'^{-1}$ has small support (support < 0.1n, say).)