On the representation of lattices by subgroup lattices

Abstract.

Whitman's condition in a lattice L means that, for any elements $a, b, c, d \in L, a \wedge b \leq c \vee d$ implies either $a \ wedge b \leq c$ or $a\wedge b \leq d$, or $a \geq c \vee d$, or $\leq bc \vee d$. We prove that any lattice satisfying Whitman’s condition can be embedded in the subgroup lattice of a free group of an arbitrary non-soluble group variety. Some interesting corollaries (both on embeddings in lattices of subgroups and others) are examined.