Two inverse results

Let A be a finite subset of a group G ₀ with |A ^?1 A|≤2|A?2. We show that there are an element α∈A and a non-null proper subgroup H of G such that one of the following holds:

• x ^?1 Hy?A ^?1 A, for all x,y∈A not both in Hα
• x Hy ^?1?AA ^?1, for all x,y∈A not both in αH

where G is the subgroup generated by A ^?1 A. Assuming that A ^?1 A≠G and that $left| {A^{ - 1} A} right| < tfrac{{5|A|}} {3} $ , we show that there are a normal subgroup K of G and a subgroup H with K?H?A ^?1 A and 2|K|≥|H| such that $A^{ - 1} AK = KA^{ - 1} A = A^{ - 1} Aand6|K| geqslant |A^{ - 1} A| = 3|H|$ .