For a $p$-group of order $p^n$, it is known that the $2$-nilpotent multiplier equal to $|\mathcal{M}^{(2)}(G)|=p^{\f12n(n-1)(n-2)+3-s_2(G)}$ for an integer $s_2(G)$. In this article, we characterize all non abelian $p$-groups satisfying in $s_2(G)\in\{1,2,3\}.

Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall: if H is a normal subgroup of a group G such that H and G/H ′ are (locally finite)-by-nilpotent, then G is (l...

Let L be an n-dimensional non-abelian nilpotent Lie algebra and $$ s(L)=\frac{1}{2}(n-1)(n-2)+1-\dim {\mathcal {M}}(L) where is the Schur multiplier of a L. The structures algebras when s(L)\in \lbrace 0,1,2,3,4,5\rbrace are determined. In this paper, we classify all s(L)=6,7.