Mathematics & Physics 2015, 8(1), 22-27 УДК 519.41/47 Some Minimal Conditions in Certain Extremely Large Classes of Groups Nikolai S.Chernikov∗ Institute of Mathematics National Academy of Sciences of Ukraine Tereschenkivska, 3, Kyiv-4, 01601 Ukraine Received 10.10.2014, received in revised form 10.11.2014, accepted 26.12.2014 Let L (respectively T) be the minimal local in the sense of D.Robinson class of groups, containing the class of weakly graded (respectively primitive graded) groups and closed with respect to forming subgroups and series. <...> In the present paper, we completely describe: the L-groups with the minimal conditions for non-abelian subgroups and for non-abelian non-normal subgroups; the T-groups with the minimal conditions for (all) subgroups and for non-normal subgroups. <...> By the way, we establish that every IHgroup, belonging to L, is solvable. <...> Keywords: local classes of groups; minimal conditions; non-abelian, Chernikov, Artinian, Dedekind, IH-groups; weakly, locally, binary, primitive graded groups. 1. <...> Some preliminary data In the present paper, the author continues his investigations [1–5]. <...> Remind that the class X of groups is called local (in our sense), if it includes every group that has a local system of subgroups belonging to X or, in the other words, a local system of X-subgroups (see [1]). <...> The class X of groups will be called local in the sense of D.Robinson, if it includes every group G such that for any finite set F of elements of G, there exists some X-subgroup S of G, containing F. (In connection with this definition, see [6, p. 93].) The following useful elementary lemmas hold. <...> Assume that some local class X of groups is closed with respect to forming subgroups. <...> Since < F >⊆ S ∈ X, < F >∈ X. Thus all finitely generated subgroups of G form its local system of X-subgroups. <...> Consequently, G ∈ X. P Remind that the class X of groups is closed with respect to forming series, if every group, having a series with X-factors, belongs to X. Sometimes a series is also called a generalized normal system (S. N.Chernikov). <...> Let X and Y be respectively the minimal local and the minimal local in the sense of D.Robinson classes <...>