Groups with Prescribed Quotient Groups and Associated Module Theory

Kurdachenko, Leonid A.; Otal, J; Subbotin, Igor Ya.

doi:10.1142/9789812778291

Cited by 23 publications

(23 citation statements)

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“…J embeds in Q , and we let U be the ideal of Q generated by L. By the above, U is a proper non-zero ideal of Q , and it follows that Q (Q /U) is finite, and hence that J/L ∼ = J has finite 0-rank. So J/L has a finitely generated subgroup C such that (J/L)/C is periodic, but π((J/L)/C ) is finite, by Corollary 1.8 of [8] (for example), and this gives the result.…”

Section: Lemma 33mentioning

confidence: 76%

“…Consider Ω P 1 (A) as a vector space over the field J/P. Again by Corollary 2.2 we have that J/P (Ω P 1 (A)) is finite, and this implies that A is artinian as a J-module, as required (see, for example, Lemma 5.6 of [8]). If A is a -group for some prime then we have from the above that B := { ∈ A : = 1} satisfies − , and it follows from Lemma 3.3 of [5] that A satisfies − .…”

Section: Lemma 32mentioning

confidence: 92%

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Groups with small deviation for non-subnormal subgroups

Kurdachenko

Smith

2009

Open Mathematics

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Abstract:We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal. MSC:20E15, 20F16, 20F19

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Section: Lemma 33mentioning

confidence: 76%

Section: Lemma 32mentioning

confidence: 92%

Groups with small deviation for non-subnormal subgroups

Kurdachenko

Smith

2009

Open Mathematics

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“…The finiteness of A implies that G/C G (A) is also finite and hence, G/C G (A) is nilpotent. Being nilpotent and finite, it is a cyclic -group (see, for example, [7,Theorem 3.1]). …”

Section: Let G Be a Group Whose Non-abelian Subgroups Are Subnormal mentioning

confidence: 99%

On the structure of groups whose non-abelian subgroups are subnormal

Kurdachenko¹,

Atlıhan

Semko³

2014

Open Mathematics

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“…For each 1 ≤ j ≤ n−1, let T j+1 /D j+1 be the periodic part of G/D j+1 . Thus T j+1 /D j+1 is a p ′ -group (see [KOS,Theorem 3.2]). Moreover, since G/T j+1 is a torsionfree abelian-by-finite locally nilpotent group, G/T j+1 is abelian.…”

Section: Locally Nilpotent Linear Groups With Wmin-icd or Wmax-icdmentioning

confidence: 99%

Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension

Kurdachenko¹,

Escolano²,

Otal³

2008

Publicacions Matemàtiques

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Let V be a vector space over a field F . If G ≤ GL(V, F ), the central dimension of G is the F -dimension of the vector space V /C V (G). In [DEK] and [KS], soluble linear groups in which the set L icd (G) of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally radical linear groups in which L icd (G) satisfies one of the weak chain conditions (the weak minimal condition or the weak maximal condition) have been characterized. In this paper, we begin the study of the non-periodic case by describing locally nilpotent linear groups in which L icd (G) satisfies one of the two weak chain conditions.

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Groups with Prescribed Quotient Groups and Associated Module Theory

Cited by 23 publications

References 0 publications

Groups with small deviation for non-subnormal subgroups

Groups with small deviation for non-subnormal subgroups

On the structure of groups whose non-abelian subgroups are subnormal

Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension

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