We prove that G 2(2m) and 3 D 4(23m) are automorphism groups of regular polytopes of rank n = 4, 5, except for the group 3 D 4(23m) when n = 5 and m is a power of 3. The minimal number of generating conjugate involutions whose product equals 1 is found, and the existence of new regular polytopes of rank 3 with these automorphism groups is established.
We study element orders in the graph automorphism extensions of finite linear and unitary groups in characteristic two.
We study the complexity of the existence problem of a ∀-sentence (∃-sentence) equivalent to a given sentence in signatures containing only unary symbols. We prove that if the signature consists of an at most one function symbol and an arbitrary number of predicate symbols, then both these problems are computable. If the signature contains at least two function symbols and arbitrarily many predicate symbols, we establish that each of these problems is an m-complete Σ_1^0 -set.
Some conditions are given under which the union of an increasing chain of Jónsson algebras is again a Jónsson algebra. Conditions are found under which there exists a countable increasing chain of Jónsson groups whose union is not a Jónsson group.
We obtain a criterion for decidable categoricity of injective unars. Exact complexity estimates are established for the index sets of the classes of strongly constructivizable injective unars and decidably categorical injective unars. It is shown that the index set of decidably categorical structures in a signature consisting of two unary functional symbols is Σ_ω +2^0 -complete.
