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Доповіді НАН України. – 2008. – N 1. – С. 14–16.
Про застосування деяких понять теорії кілець для вивчення впливу систем підгруп групи
М.М. Пискун
Abstract
Let A be a partially ordered set. For a, b О A,
we put [a,b]= {x ОA | a ≤ x ≤ b}. The deviation of A, denoted
as dev(A), is defined by the following rule.
If A is trivial, then we put dev(A) = – ∞.
If A is not trivial but satisfies the minimal condition, then
dev(A) = 0. For a general ordinal α, we define
dev(A) = α provided dev(A) №
β < α and, in any descending chain a1 ≥ a2
≥ ј ≥ an ≥ ј of elements of
A, all but finitely many of the closed intervals [an,an+1] have
deviation less than α. 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
Lnon-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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