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Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$, there exists a smallest ideal $I$ of $L$ such that $L/I\in 𝒩$. This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$. In this paper, we define the subalgebra $S\left(L\right)={\bigcap }_{H\le L}{I}_L\left(H^{𝒩}\right)$. Set $S_0\left(L\right)=0$. Define $S_{i+1}\left(L\right)/S_i\left(L\right)=S(L/S_i\left(L\right))$ for $i\ge 1$. By $S_{\infty }\left(L\right)$ denote the terminal term of the ascending series. It is proved that $L=S_{\infty }\left(L\right)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a Lie algebra...

In this paper, we give a complete classification of the finite groups G whose second maximal subgroups are cyclic

We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than $\frac{29}{4}$ then $G$ is solvable or $G/F\left(G\right)\cong A_5$. This generalizes an earlier result by the third author.