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Let L be an n-dimensional non-abelian nilpotent Lie algebra and $$s\left(L\right)=\frac{1}{2}(n-1)(n-2)+1-dimM\left(L\right)$$ where M(L) is the Schur multiplier of L. In [Niroomand P., Russo F., A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra (in press)] it has been shown that s(L) ≥ 0 and the structure of all nilpotent Lie algebras has been determined when s(L) = 0. In the present paper, we will characterize all finite dimensional nilpotent Lie algebras with s(L) = 1; 2.

We conduct an in-depth investigation into the structure of the Bogomolov multiplier for groups of order $p^7$ $(p>2)$ and exponent $p$. We present a comprehensive classification of these groups, identifying those with nontrivial Bogomolov multipliers and distinguishing them from groups with trivial multipliers. Our analysis not only clarifies the conditions under which the Bogomolov multiplier is nontrivial but also refines existing computational methods, enhancing the process of determining...