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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
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Abstract

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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

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