Mathematics & Physics 2016, 9(3), 393–400 УДК 512.54 On Normal Closures of Involutions in the Group of Limited Permutations Yuri S.Tarasov∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.03.2016, received in revised form 08.05.2016, accepted 20.06.2016 We study the group G = Lim(N) of limited permutations of a set N of all natural numbers. <...> Found the link between the dispersion subsets of a set N and normal subgroups of G. Keywords: group, limited permutation, dispersion set, normal subgroup, involution. <...> Introduction Let N be the set of all natural numbers, Z be the set of all integers, M is any of these sets. <...> The S(M) will denote the group of all permutations of the set M. Definition 1. <...> Thus set Lim(M) = {x|x ∈ S(M),w(x) < ∞} form a group, which is a natural extension of a locally finite group Fin(M) of all finitary permutations of the set M, i.e. such permutation y ∈ S(M), for which the set {α|α ∈M,αy ̸= α} is finite. <...> In the work of N.M.Suchkov [1] an example of the mixed group H = AB was first built, where A,B is periodic (and even locally-finite) subgroups. <...> Then in [2, 3] it was found that H = ⟨g |g ∈ Lim(Z), |g| < ∞⟩, any countable free group and Aleshin 2-group isomorphically embeddable into the group H and Lim(Z) = H ⟨d⟩, where d-shift, αd = α+1 for any α ∈ Z. ∗gigtorus@yandex.ru ⃝ Siberian Federal University. <...> All rights reserved c – 393 – Yuri S.Tarasov On Normal Closures of Involutions in the Group of Limited Permutations In their work [4] N.M.Suchkov and N. G. Suchkova proved the factorization of the whole group Lim(N) by two locally-finite subgroups and it is shown that the group Lim(M) is generated by the permutations x ∈ S(M), for which w(x) = 1. <...> These generators are either involutions, in which decomposition into independent cycles only transpositions of the (αα + 1), α ∈ M or M = Z and x ∈ {d, d−1}. <...> The relation between groups Lim(N) and Lim <...>