Let V be a vector space over a field F . If G ≤ GL V F , the central dimension of G is the F -dimension of the vector space V/C V G . In Dixon et al. (2004) and Kurdachenko and Subbotin (2006), soluble linear groups in which the set icd G of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. In this article we study periodic locally radical linear groups, in which the set icd G satisfies one of the weak chain conditions: the weak minimal condition or the weak maximal condition.
Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[\cdot,\cdot]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity $[[a,b],c]=[a,[b,c]]-[b,[a, c]]$ for all $a,b,c\in L$. This paper is a brief review of some current results, which related to finite-dimensional and infinite-dimensional Leibniz algebras.
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