By several postulates we introduce a new class of algebraic lattices, in which a main role is played by so called normal elements. A model of these lattices are weak-congruence lattices of groups, so that normal elements correspond to normal subgroups of subgroups. We prove that in this framework many basic structural properties of groups turn out to be lattice-theoretic. Consequently, we give necessary and sufficient conditions under which a group is Hamiltonian, Dedekind, abelian, solvable, supersolvable, metabelian, finite nilpotent. These conditions are given as lattice-theoretic properties of a lattice with normal elements.

中文翻译：

---

具有普通元素的格子

通过几个假设，我们引入了一类新的代数格，其中所谓的正规元素起着主要作用。这些格的模型是群的弱同余格，因此正常元素对应于子群的正常子群。我们证明，在这个框架中，群的许多基本结构性质都是格论的。因此，我们给出了一个群是哈密顿群、戴德金群、阿贝尔群、可解群、超可解群、元贝尔群、有限幂零群的充要条件。这些条件作为具有普通元素的晶格的晶格理论特性给出。