Про застосування деяких понять теорії кілець для вивчення впливу систем підгруп групи

Пискун, М. М.

Let A be a partially ordered set. For <$E a,~b~symbol <174>~A>, we put <$E [a,~b]~=~left { x~symbol <174>~A~|~a~symbol Г~x~symbol Г~b right }>. The deviation of A, denoted as dev(A), is defined by the following rule. If A is trivial, then we put <$E roman dev (A)~=~- inf>. If A is not trivial but satisfies the minimal condition, then dev(A) = 0. For a general ordinal <$E alpha>, we define <$E roman dev (A)~=~alpha> provided <$E roman dev (A)~symbol Щ~beta~<<~alpha> and, in any descending chain <$E a sub 1~symbol У~a sub 2~symbol У~cdot cdot cdot~symbol У~a sub n~symbol У~cdot cdot cdot> of elements of A, all but finitely many of the closed intervals <$E [a sub n ,~a sub n+1 ]> have deviation less than <$E alpha>. Let G be a group and let S be some family of subgroups of G. Then S is partially ordered by inclusion. If a partially ordered set S has a deviation, then we will say that a family S has the Krull dimension. In this paper, we study the groups, in which the family <$E L sub roman non-nn (G)> of all non nearly normal subgroups has the Krull dimension. A subgroup H of the group G is said to be nearly normal, if H has finite index in its normal closure.


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