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B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

Published online by Cambridge University Press:  17 April 2009

A. Abdollahi ,
A. Azad ,
A. Mohammadi Hassanabadi  and
M. Zarrin
A. Abdollahi
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: a.abdollahi@math.ui.ac.ir
A. Azad
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: a-azad@sci.ui.ac.ir
A. Mohammadi Hassanabadi
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: aamohaha@sci.ui.ac.ir
M. Zarrin
Affiliation:
Department of MathematicsUniversity of IsfahanIsfahan 81746-73441Iran e-mail: m.zarrin@math.ui.ac.ir
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This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 1 , August 2006 , pp. 121 - 132
Copyright
Copyright © Australian Mathematical Society 2006

References

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