Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T00:08:26.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Small squaring and cubing properties for finite groups

Published online by Cambridge University Press:  17 April 2009

Ya. G. Berkovich ,
G.A. Freiman  and
Cheryl E. Praeger
Ya. G. Berkovich
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
G.A. Freiman
Affiliation:
School of Mathematical, Sciences Raymond and Beverley Sackler, Faculty of Exact Sciences, Tel-Aviv University, Ramat - Aviv 69978, Israel
Cheryl E. Praeger
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands WA 6009, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A group G is said to have the small squaring property on k-sets if |K2| < k2 for all k-element subsets K of G, and is said to have the small cubing property on k-sets if |K3| < k3 for all k-element subsets K. It is shown that a finite nonabelian group with the small squaring property on 3-sets is either a 2-group or is of the form TP with T a normal abelian odd order subgroup and P a nontrivial 2-group such that Q = Cp(T) has index 2 in P and P inverts T. Moreover either P is abelian and Q is elementary abelian, or Q is abelian and each element of P − Q inverts Q. Conversely each group of the form TP as above has the small squaring property on 3-sets. As for the nonabelian 2-groups with the small squaring property on 3-sets, those of exponent greater then 4 are classified and the examples are similar to dihedral or generalised quaternion groups. The remaining classification problem of exponent 4 nonabelian examples is not complete, but these examples are shown to have derived length 2, centre of exponent at most 4, and derived quotient of exponent at most 4. Further it is shown that a nonabelian group G satisfies |K2| < 7 for all 3-element subsets K if and only if G = S3. Also groups with the small cubing property on 2-sets are investigated.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 3 , December 1991 , pp. 429 - 450
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Berkovich, Ya. G., ‘On p−groups of finite order’, Siberian Math. J. 9 (1968), 1284–1306. (in Russian; English translation).CrossRefGoogle Scholar
[2]Berkovich, Ya. G., ‘Generalization of theorems of P. Hall and Blackburn and an application to non-regular p−groups’, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 800–831. (in Russian; English translation).Google Scholar
[3]Berkovich, Ya. G. and Freiman, G.A., ‘Group structure and cardinality of powers of its small subsets’, (unpublished manuscript, 1981).Google Scholar
[4]Brailovsky, L.V. and Freiman, G.A., ‘On two-element subsets in groups’, Ann. New York Acad. Sci. 373 (1981), 183–190.CrossRefGoogle Scholar
[5]Brailovsky, L.V. and Freiman, G.A., ‘On a product of finite subsets in a torsion-free group’, J. Algebra 130 (1990), 462–476.CrossRefGoogle Scholar
[6]Brailovsky, L.V. and Freiman, G.A., ‘Groups with small cardinality of their two-element subsets’, Ann. New York Acad. Sci 410 (1983), 75–82.CrossRefGoogle Scholar
[7]Freiman, G.A., Foundations of a structural theory of set addition, (in Russian) (Kazan, 1966). Translations of Mathematical Monographs 37, (English translation) (Amer. Math. Soc., Providence, R.I., 1973).Google Scholar
[8]Freiman, G.A., ‘Groups and the inverse problems of the additive set theory’, in Number-theoretic investigations on the Markov spectrum and the structure theory of set addition, (in Russian), pp. 175–183 (Kalinin University, Moscow, 1973).Google Scholar
[9]Freiman, G.A., ‘On two- and three-element subsets of groups’, Aequationes Math. 22 (1981), 140–152.CrossRefGoogle Scholar
[10]Freiman, G.A. and Schein, B.M., ‘Group and semigroup theoretic considerations inspired by inverse problems of the additive number theory’, in Lecture Notes in Math. 1320, pp. 121–140 (Springer-Verlag, Berlin-Heidelberg-New York, 1988).Google Scholar
[11]Golfand, J.A., ‘On groups all of whose subgroups are special’, Dokl. Akad. Nauk SSSR) 60, No. 8 (1948), 1313–1315.Google Scholar
[12]Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
[13]Huppert, B., Endliche gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[14]Kemperman, J.H.B., ‘On complexes in a semigroup’, Indag. Math. 18 (1956), 247–254.CrossRefGoogle Scholar
[15]Nekrasov, K.G. and Berkovich, Ya.G., ‘A necessary and sufficient condition for the cyclicity of the Frattini subgroup of a finite p−group, Voprosyteorii Grup, Gomologicheskoi algebry, Yaroslarl’, pp. 55–57, (1985).Google Scholar
[16]Nekrasov, K.G., private communication, 1987.Google Scholar
[17]Neumann, P.M., private communication, 1989.Google Scholar
[18]Robinson, D.J.S., A course in the theory of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar