Non abelian tensor square of non abelian prime power groups

For every $p$-group of order $p^n$ with the derived subgroup of order $p^m$, Rocco in \cite{roc} has shown that the order of tensor square of $G$ is at most $p^{n(n-m)}$. In the present paper not only we improve his bound for non-abelian $p$-groups but also we describe the structure of all non-abelian $p$-groups when the bound is attained for a special case. Moreover, our results give as well an upper bound for the order of $\pi_3(SK(G, 1))$.Comment: enriched with contributions of F.G. Russ

Structure of nilpotent Lie algebra by its multiplier

For a finite dimensional Lie algebra $L$, it is known that s(L)=\f{1}{2}(n-1)(n-2)+1-\mathrm{dim} M(L) is non negative. Moreover, the structure of all finite nilpotent Lie algebras is characterized when $s(L)=0,1$ in \cite{ni,ni4}. In this paper, we intend to characterize all nilpotent Lie algebra while $s(L)=2.

Characterization of finite dimensional nilpotent Lie algebras by the dimension of their Schur multipliers, s(L)=5s(L)=5

It is known that the dimension of the Schur multiplier of a non-abelian nilpotent Lie algebra $L$ of dimension $n$ is equal to $\frac{1}{2}(n-1)(n-2)+1-s(L)$ for some $s(L)\geq0$. The structure of all nilpotent Lie algebras has been given for $s(L) \leq 4$ in several papers. Here, we are going to give the structure of all non-abelian nilpotent Lie algebras for $s(L)=5$

Capability of Nilpotent Lie algebras with small derived Subalgebra

In this paper, we classify all capable nilpotent Lie algebras with derived subalgebra of dimension at most 1.Comment: To appear in J. Algebra with new contributions of F.G. Russ