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О стабильных элементах в свободных нильпотентных группах ранга два.

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1.
512.54
doi: 10.18097/1994–0866–2015–0–9–3–6
©
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, 1, e-mail: annkow@mail.ru
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STABLE ELEMENTS IN FREE NILPOTENT GROUPS OF RANK TWO
Anna I. Kovyrshina
PhD, A/Professor, Pedagogical Institute of Irkutsk State University
1 Karla Marksa st., Irkutsk 664003, Russia
The article presents a complete description of the elements of free nilpotent groups of rank two and
stage 8 that remain unchanged under the action of any automorphism of the group. Existence of nontrivial stable elements in such groups is known, but their number was not found. We prove a theorem
on the uniqueness (up to its powers) of nontrivial stable element in free nilpotent groups of rank two
and stage 8.
Keywords: automorphisms of groups, fixed points.
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//
:
.
2.
.–
, 1998. – . 5.
.
.
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9/2015
//
. – 2010. – 4(58). – . 20–23.
3.
. .
//
.
.
. – 2010. – . 3,
4. – . 48–57.
4. Burrow M. D. Invariants of free Lie rings // Communications on pure and applied mathematics. – 1958. – No. 11. – Pp. 419–431.
5. Burrow M. D. The enumeration of Lie invariants // Communications on pure and applied mathematics. – 1967. – No. 20. – Pp. 401–411.
6. Formanek E. Fixed points and centers of automorphism groups of free nilpotent groups // Communications in algebra. 2002. No. 30. Pp. 1033–1038.
7.
.,
.,
.
. – .:
, 1974. –
455 .
8. Papistas A. A note on fixed points of certain relatively free nilpotent groups //Communications
in algebra. 2001. No. 29. Pp. 4693–4699.
9. Wever F. Ueber Invarianten in Lieschen Ringen // Mathematische Annalen. – 1949. – No. 120. –
Pp. 563–580.
References
1. Bludov V. V. Nepodvizhnye tochki otnositel'no vsekh avtomorfizmov v svobodnykh nil'potentnykh gruppakh [Fixed points with respect to all automorphisms in free nilpotent groups]. Tretii Sibirskii kongress po prikladnoi i industrial'noi matematike – Third Siberian Congress on Industrial and
Applied Mathematics. Part 5. Novosibirsk, 1998.
2. Kovyrshina A. I. Stabil'nye elementy v svobodnykh nil'potentnykh gruppakh ranga tri [Fixed
points in free nilpotent groups of rank three]. Vestnik Omskogo universiteta – Bulletin of Omsk University. 2010. No. 4 (58). Pp. 20–23.
3. Kovyrshina A. I. Stabil'nye elementy v svobodnykh nil'potentnykh gruppakh ranga dva [Stable
elements of free nilpotent of rank two]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya:
Matematika. Proceedings of Irkutsk State University. Series: Mathematics, 2010. No. 4. Pp. 48–57.
4. Burrow M. D. Invariants of free Lie rings. Communications on pure and applied mathematics.
1958. No. 11. Pp. 419–431.
5. Burrow M. D. The enumeration of Lie invariants. Communications on pure and applied mathematics. 1967. No. 20. Pp. 401–411.
6. Formanek E. Fixed points and centers of automorphism groups of free nilpotent groups. Communications in algebra. 2002. No. 30. Pp. 1033–1038.
7. Magnus W., Karras A., Solitar D. Combinatorial Group Theory: Presentations of Groups in
Terms of Generators and Relations. New York-London-Sydney: John Wiley and Sons, Inc., 1966. 444
p.
8. Papistas A. A note on fixed points of certain relatively free nilpotent groups. Communications in
algebra. 2001. No. 29. Pp. 4693–4699.
9. Wever F. Ueber Invarianten in Lieschen Ringen. Mathematische Annalen. 1949. No. 120.
Pp. 563–580. (Ger.)
6
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