A subgroup H of a group G is said to be quasinormal if HX = XH for all subgroups X of G. In this article groups are characterized for which the partially ordered set of quasinormal subgroups is decomposable.
A group G is called metamodular if for each subgroup H of G either the subgroup lattice 𝔏(H) is modular or H is a modular element of the lattice 𝔏(G). Metamodular groups appear as the natural lattice analogues of groups in which every non-abelian subgroup is normal; these latter groups have been studied by Romalis and Sesekin, and here their results are extended to metamodular groups.
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