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GROUPS WITH COMMUTING POWERS

Part of: Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  13 March 2009

ROLF BRANDL
ROLF BRANDL*
Affiliation:
Mathematisches Institut, Am Hubland 12, 97074 Würzburg, Germany
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Abstract

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A group G satisfies the second Engel condition [X,Y,Y ]=1 if and only if x commutes with xy, for all x,y∈G. This paper considers the generalization of this condition to groups G such that, for fixed positive integers r and s, xr commutes with (xs)y for all x,y∈G. Various general bounds are proved for the structure of groups in the corresponding variety, defined by the law [Xr,(Xs)Y]=1.

Keywords

two-EngelBell groupsvariety of groups

MSC classification

Secondary: 20F19: Generalizations of solvable and nilpotent groups 20F12: Commutator calculus 20F45: Engel conditions 20D60: Arithmetic and combinatorial problems 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 343 - 351
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Brandl, R., ‘Zur Theorie der untergruppenabgeschlossenen Formationen: Endliche Varietäten’, J. Algebra 73 (1981), 1–22.Google Scholar
[2] Brandl, R. and Kappe, L. C., ‘On n-Bell groups’, Comm. Algebra 17 (1989), 787–807.CrossRefGoogle Scholar
[3] Bryant, R. M. and Powell, M. B., ‘Two-variable laws for PSL(2,5)’, J. Aust. Math. Soc. 10 (1969), 499–502.Google Scholar
[4] Carter, R., Fischer, B. and Hawkes, T. O., ‘Extreme classes of finite groups’, J. Algebra 9 (1968), 285–313.Google Scholar
[5] Huppert, B., Endliche Gruppen, Vol. I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[6] Huppert, B. and Blackburn, N., Finite Groups, Vol. II (Springer, Berlin, 1982).Google Scholar
[7] Kappe, L. C., ‘On n-Levi groups’, Arch. Math. 47 (1986), 198–210.Google Scholar
[8] Vaughan-Lee, M., ‘Derived lengths of Burnside groups of exponent 4’, Q. J. Math. (Oxford) II. Ser. 30 (1979), 495–504.Google Scholar
[9] Zelmanov, E. I., ‘Solution of the restricted Burnside problem for groups of odd order’, Izv. Akad. Nauk SSSR Ser. Mat. 54(1) (1990), 42–59.Google Scholar
[10] Zelmanov, E. I., ‘Solution of the restricted Burnside problem for 2-groups’, Mat. Sb. 182(4) (1991), 568–592.Google Scholar