Given an abstract group G, we study the function a b n ( G ) : = sup | G : H | ≤ n | H / [ H , H ] |
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We show the existence of an absolute constant α> 0 α> 0 such that, for every k ≥ 3 k≥ 3,
G:= Sym (k) G:= Sym (k), and for every H\leqslant GH⩽ G of index at least 3, one has| H/H' …

Suppose that w= w⁢(x 1,…, xn) is a word, ie an element of the free group F=⟨ x 1,…, xn⟩.
The verbal subgroup w⁢(G) of a group 𝐺 is the subgroup generated by the set {w⁢(x 1 …