We consider a lattice generated by three elements, two of which are semi-normal. We have proved it is infinite and have also found an additional condition for the lattice to be modular. As a result, it is proved that the sublattice generated in the subgroup lattice by two normal subgroups and any subgroup is modular.
Lattice Semi-normal element Defining relation Finite lattice Free lattice
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Gein, A.G., Shushpanov, M.P.: Sufficient conditions for the modularity of the lattice generated by elements with properties of modular type. Sib. Math. J. 56(4), 631–636 (2015)MathSciNetCrossRefMATHGoogle Scholar