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9.62 (1984)

Solved

In any group $G$, the cosets of all of its normal subgroups together with the empty set form the block lattice $C(G)$ with respect to inclusion, which, for infinite $|G|$, is subdirectly irreducible and, for finite $|G| \geqslant 3$, is even simple (D. M. Smirnov, A. V. Reibol’d, Algebra and Logic, 23, no. 6 (1984), 459–470). How large is the class of such lattices? Is every finite lattice embeddable in the lattice $C(G)$ for some finite group $G$?

Contributor: D. M. Smirnov

Progress

For each $g \in G$ the filter $\{x \in C(G) \mid g \in x\}$ is modular (and even arguesian); the variety generated by the block lattices of groups does not contain all finite lattices (D. M. Smirnov, Siberian Math. J., 33, no. 4 (1992), 663–668).

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