Counterexamples to two conjectures in the Kourovka notebook

Abstract

2.6. In an N R-group, the set of generalized central elements coincides with the nilkernel. Is the converse true, that is, must a group be an N R-group if the set of generalized central elements coincides with the nil-kernel?V. G. Vilyatser 2.9. Do there exist regular associative operations on the class of groups satisfying the weakened Mal'cev condition (that is, monomorphisms of the factors of an arbitrary product can be glued together, generally speaking, into a homomorphism of the whole product), but not satisfying the analogous condition for epimorphisms of the factors?O. N. Golovin 2.22. a) An abstract group-theoretic property Σ is said to be radical (in our sense) if, in any group G, the subgroup Σ(G) generated by all normal Σ-subgroups is a Σsubgroup itself (called the Σ-radical of G). A radical property Σ is said to be strongly radical if, for any group G, the factor-group G/Σ(G) contains no non-trivial normal Σ-subgroups. Is the property RN radical? strongly radical? Sh. S. Kemkhadze 2.24. Are Engel torsion-free groups orderable? A. I. Kokorin 2.25. a) (L. Fuchs). Describe the groups which are linearly orderable in only finitely many ways. A. I. Kokorin 2.26. (L. Fuchs). Characterize as abstract groups the multiplicative groups of orderable skew fields. A. I. Kokorin 2.28. Can every orderable group be embedded in a pro-orderable group? (See 1.35for the definition of pro-orderable groups.) A. I. Kokorin 2.40. c) The I-theory (Q-theory) of a class K of universal algebras is the totality of all identities (quasi-identities) that are valid on all algebras in K. Does there exist a finitely axiomatizable variety (3) of Lie rings (i) whose I-theory is non-decidable? Remark: it is not difficult to find a recursively axiomatizable variety of semigroups with identity whose I-theory is non-recursive (see also A. I. Mal'cev, Mat. Sbornik, 69, no. 1 (1966), 3-12 (Russian)). A. I. Mal'cev 2nd Issue (1967) 2.42. What is the structure of the groupoid of quasivarieties a) of all semigroups? b) of all rings?

On 2-closures of rank 3 groups

“…It is important to stress that there are cases when a one-dimensional rank 3 group has a nonsolvable 2-closure; see [28] for such an example of degree 2 6 .…”

The $3$-closure of a solvable permutation group is solvable

“…The case of 2-closure is more interesting. The 2-closure of every (solvable) 2-transitive group G ≤ Sym(Ω) is Sym(Ω); other examples of solvable G and nonsolvable G (2) appear in [18]. But, as shown by Wielandt [20], each of the classes of finite p-groups and groups of odd order is closed with respect to taking the 2-closure.…”

Two-closure of supersolvable permutation group in polynomial time

“…A challenging problem is to find the 2-closure G of a solvable group G. The main obstacle here is that among the composition factors of G, simple groups other than cyclic and alternating can occur: the examples have recently been found in [17]. This effect does not appear for the supersolvable groups, this follows from the theorem below, obtained as a byproduct of a technique used to prove Theorem 1.1.…”