  The Brown-Erd\H{o}s-S\'{o}s conjecture, one of the central conjectures in
extremal combinatorics, states that for any integer $m\geq 6,$ if a 3-uniform
hypergraph on $n$ vertices contains no $m$ vertices spanning at least $m-3$
edges, then the number of edges is $o(n^2).$ We prove the conjecture for triple
systems coming from finite abelian groups.
