Vol 57, No 3 (2018)

Let ???? be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal ????-subgroup if there exists an isomorpic embedding ϕ: G ↪ G^* of the group G into some finite group G^* under which G^ϕ is subnormal in G^* and H^ϕ = K ∩G^ϕ for some maximal ????-subgroup K of G^*. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal ????-subgroups are conjugate in a finite group G in which all maximal ????-subgroups are conjugate? This question strengthens Wielandt’s known problem of closedness for the class of -groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group.

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. We study lattice isomorphisms of finite commutative rings with identity. The objective is to specify sufficient conditions subject to which rings under lattice homomorphisms preserve the following properties: to be a commutative ring, to be a ring with identity, to be decomposable into a direct sum of ideals. We look into the question about the projective image of the Jacobson radical of a ring. In the first part, the previously obtained results on projections of finite commutative semiprime rings are supplemented with new information. Lattice isomorphisms of finite commutative rings decomposable into direct sums of fields and nilpotent ideals are taken up in the second part. Rings definable by their subring lattices are exemplified. Projections of finite commutative rings decomposable into direct sums of Galois rings and nilpotent ideals are considered in the third part. It is proved that the presence in a ring of a direct summand definable by its subring lattice (i.e., the Galois ring GR(pⁿ,m), where n > 1 and m > 1) leads to strong connections between the properties of R and R′.

Necessary and sufficient conditions are stated for an arbitrary theory to be an elementary theory for a class of its existentially closed models. Conditions are given under which some existentially closed model simultaneously realizes one maximal existential type and omits another. We also prove a theorem on a prime existentially closed model over a maximal existential type. Considerable complexity of existentially closed structures and their theories was noted by A. Macintyre. Therefore, the examples of existentially closed companions having any finite or countable number of pairwise non elementarily equivalent existentially closed models constructed here are of interest.

It is known that a modular 3-generated lattice is always finite and contains at most 28 elements. Lattices generated by three elements with certain modularity properties may no longer be modular but nevertheless remain finite. It is shown that a 3-generated lattice among generating elements of which one is seminormal and another is coseminormal is finite and contains at most 45 elements. This estimate is stated to be sharp.