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Antifinitary linear groups. (English) Zbl 1145.20025

It is not clear what the opposite notion to finitary for linear groups should be. As an alternative to the earlier suggestions of cofinitary and contrafinitary [e.g., the reviewer, Monatsh. Math. 121, No. 4, 391-397 (1996; Zbl 0848.20042)], the authors suggest one of their own, which they name antifinitary linear group.
Let \(V\) be a vector space over some field \(F\) and suppose \(G\) is a subgroup of \(\text{GL}(V,F)=\operatorname{Aut}_FV\). If \(H\) is any subgroup of \(G\) let \(\mathbf h\) denote its augmentation ideal. The authors say that \(G\) is antifinitary if every proper subgroup \(H\) of \(G\) with \(V\mathbf h\) of infinite dimension over \(F\) is finitely generated. Recall that \(G\) is finitary if for every finitely generated subgroup \(H\) of \(G\), the space \(V\mathbf h\) has finite dimension over \(F\).
The authors point out that every group with all its proper subgroups finitely generated (e.g., a Tarski monster) is isomorphic to an antifinitary linear group, so little can be expected without further hypotheses. Thus the authors focus here certain generalized soluble groups.
Suppose \(G\) is a non finitely-generated, locally hyper (locally-nilpotent by locally-finite) antifinitary subgroup of \(\text{GL}(V,F)\). The finitary radical \(R\) of \(G\) is the set of all \(g\) in \(G\) with \(\dim_FV(g-1)\) finite. The authors prove that if \(G/R\) is not finitely generated, then \(G\) is a Prüfer \(p\)-group for some prime \(p\), while if \(G/R\) is finitely generated then \(G\) is an extension of a divisible Abelian Chernikov group by a cyclic group of prime power order, with a very specific structure (Theorem A).
The authors’ second main theorem (Theorem B) describes the structure of a finitely generated, hyper locally nilpotent, antifinitary subgroup of \(\text{GL}(V,F)\) with \(\dim_FV\mathbf g\) infinite. It is too involved to state here, but as a taster it implies that \(G\) is nilpotent by polycyclic.

MSC:

20H20 Other matrix groups over fields
20F22 Other classes of groups defined by subgroup chains
20F19 Generalizations of solvable and nilpotent groups

Citations:

Zbl 0848.20042
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References:

[1] DOI: 10.1016/j.jalgebra.2004.02.029 · Zbl 1055.20042 · doi:10.1016/j.jalgebra.2004.02.029
[2] DOI: 10.1017/S0305004100053834 · Zbl 0365.20047 · doi:10.1017/S0305004100053834
[3] DOI: 10.1112/jlms/s2-37.3.421 · Zbl 0619.20018 · doi:10.1112/jlms/s2-37.3.421
[4] Maltsev A. I., Math. Sb. 8 pp 405– (1940)
[5] Phillips R. E., Dordrecht pp 111– (1995)
[6] DOI: 10.1016/0021-8693(72)90058-0 · Zbl 0236.20032 · doi:10.1016/0021-8693(72)90058-0
[7] DOI: 10.1006/jabr.1995.1312 · Zbl 0837.20060 · doi:10.1006/jabr.1995.1312
[8] DOI: 10.1002/mana.19951760122 · Zbl 0842.20042 · doi:10.1002/mana.19951760122
[9] DOI: 10.1515/form.1997.9.603 · Zbl 0883.20023 · doi:10.1515/form.1997.9.603
[10] Zaitsev D. I., Ukrain. Math. J. 23 pp 652– (1971)
[11] Zaitsev D. I., Math. Inst. Kiev pp 72– (1974)
[12] Zaitsev D. I., Dokl. Akad Nauk SSSR 240 pp 257– (1978)
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