Vol 57, No 1 (2018)

We give a description of spectra of finite simple and universal groups of Lie type E₈.

A group G is said to be rigid if it contains a normal series G = G₁ > G₂ > . . . > G_m > G_m+1 = 1, whose quotients G_i/G_i+1 are Abelian and, treated as right ℤ[G/G_i]- modules, are torsion-free. A rigid group G is divisible if elements of the quotient G_i/G_i+1 are divisible by nonzero elements of the ring ℤ[G/G_i]. Every rigid group is embedded in a divisible one. Previously, it was stated that the theory ????_m of divisible m-rigid groups is complete. Here, it is proved that this theory is ω-stable. Furthermore, we describe saturated models, study elementary submodels of an arbitrary model, and find a representation for a countable saturated model in the form of a limit group in the Fraïssé system of all finitely generated m-rigid groups. Also, it is proved that the theory ????_m admits quantifier elimination down to a Boolean combination of ∀∃-formulas.

A Schur ring (an S-ring) is said to be separable if each of its algebraic isomorphisms is induced by an isomorphism. Let C_n be the cyclic group of order n. It is proved that all S-rings over groups \( D={C}_p\times {C}_{p^k} \), where p ∈ {2, 3} and k ≥ 1, are separable with respect to a class of S-rings over Abelian groups. From this statement, we deduce that a given Cayley graph over D and a given Cayley graph over an arbitrary Abelian group can be checked for isomorphism in polynomial time with respect to |D|.

A group G is saturated with groups from a set ℜ of groups if every finite subgroup of G is contained in a subgroup of G that is isomorphic to some group in ℜ. Previously [Kourovka Notebook, Quest. 14.101], the question was posed whether a periodic group saturated with finite simple groups of Lie type whose ranks are bounded in totality is itself a simple group of Lie type. A partial answer to this question is given for groups of Lie type of rank 1. We prove the following: Let a periodic group G be saturated with finite simple groups of Lie type of rank 1. Then G is isomorphic to a simple group of Lie type of rank 1 over a suitable locally finite field.