Toward the reverse decomposition of unipotents. II. The relative case

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Toward the reverse decomposition of unipotents. II. The relative case. / Вавилов, Николай Александрович.

In: Journal of Mathematical Sciences, Vol. 252, No. 6, 02.2021, p. 749-760.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{1c6de282eb134a58b5880970eeec9c4e,

title = "Toward the reverse decomposition of unipotents. II. The relative case",

abstract = "Recently, Raimund Preusser displayed very short polynomial expressions of elementary generatorsin classical groups over an arbitrary commutative ring as products of conjugates of an arbitrarymatrix and its inverse by absolute elementary matrices. In particular, this provides very shortproofs for description of normal subgroups. In 2018, the author discussed various generalizationsof these results to exceptional groups, specifically those of types E6and E7. Here, a furthervariation of Preusser{\textquoteright}s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4 ,similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1by relativeelementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J R, are obtained. Again,in particular, this allows to give very short proofs for the description of subgroups normalized byE(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).",

keywords = "General linear group, elementary subgroup, subnormal subgroup",

author = "Вавилов, {Николай Александрович}",

year = "2021",

month = feb,

doi = "10.1007/s10958-021-05195-8",

language = "English",

volume = "252",

pages = "749--760",

journal = "Journal of Mathematical Sciences",

issn = "1072-3374",

publisher = "Springer Nature",

number = "6",

}

RIS

TY - JOUR

T1 - Toward the reverse decomposition of unipotents. II. The relative case

AU - Вавилов, Николай Александрович

PY - 2021/2

Y1 - 2021/2

N2 - Recently, Raimund Preusser displayed very short polynomial expressions of elementary generatorsin classical groups over an arbitrary commutative ring as products of conjugates of an arbitrarymatrix and its inverse by absolute elementary matrices. In particular, this provides very shortproofs for description of normal subgroups. In 2018, the author discussed various generalizationsof these results to exceptional groups, specifically those of types E6and E7. Here, a furthervariation of Preusser’s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4 ,similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1by relativeelementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J R, are obtained. Again,in particular, this allows to give very short proofs for the description of subgroups normalized byE(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).

AB - Recently, Raimund Preusser displayed very short polynomial expressions of elementary generatorsin classical groups over an arbitrary commutative ring as products of conjugates of an arbitrarymatrix and its inverse by absolute elementary matrices. In particular, this provides very shortproofs for description of normal subgroups. In 2018, the author discussed various generalizationsof these results to exceptional groups, specifically those of types E6and E7. Here, a furthervariation of Preusser’s wonderful idea is presented. Namely, in the case of GL(n, R), n ≥ 4 ,similar expressions of elementary transvections as conjugates of g ∈ GL(n, R) and g−1by relativeelementary matrices x ∈ E(n, J) and then x ∈ E(n, R, J), for an ideal J R, are obtained. Again,in particular, this allows to give very short proofs for the description of subgroups normalized byE(n, J) or E(n, R, J), and thus also of subnormal subgroups in GL(n, R).

KW - General linear group

KW - elementary subgroup

KW - subnormal subgroup

UR - http://www.scopus.com/inward/record.url?scp=85099766254&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/7bed154d-1a79-3c7c-b0a9-4405babee3e3/

U2 - 10.1007/s10958-021-05195-8

DO - 10.1007/s10958-021-05195-8

M3 - Article

VL - 252

SP - 749

EP - 760

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 6

ER -

ID: 72838414