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How commutative can a non-commutative group be?

Published online by Cambridge University Press:  03 November 2016

Desmond MacHale
Desmond MacHale*
Affiliation:
Department of Mathematics, University College, Cork
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Let G denote a finite non-commutative group. What can we say about the number of ordered pairs (x,y), where x,y ϵ G, such that xy = yx?

In order to prepare ourselves to answer this question, let us examine the Cayley table of S 3, the symmetric group on three symbols. We take as generators the reflection (12) and the rotation (123). It is clear that we are examining the Cayley table for symmetry about its main diagonal.

Information

Type
Research Article
Information
The Mathematical Gazette , Volume 58 , Issue 405 , October 1974 , pp. 199 - 202
Copyright
Copyright © Mathematical Association 1974

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References

1. Joseph, K. S., Commutativity in non-abelian groups. Ph.D. thesis, University of California, Los Angeles (1969).Google Scholar
2. Gallagher, P. X., The number of conjugacy classes in a finite group, Math. Z. 118, 175–179 (1970).Google Scholar
3. MacHale, P. D., Finite groups with automorphisms inverting many elements. Ph.D. thesis, University of Keele (1972).Google Scholar