  The first open case of the Brown, Erd\H{o}s, S\'os conjecture is equivalent
to the following; For every $c>0$ there is a threshold $n_0$ so that if a
quasigroup has order $n\geq n_0$ then for every subset of triples of the form
$(a,b,ab),$ denoted by $S,$ if $|S|\geq cn^2$ then there is a seven-element
subset of the quasigroup which spans at least four triples of the selected
subset $S.$ In this paper we prove the conjecture for finite groups.
