Order Conjecture for non-commuting graph of a group

The following was conjectured by A. Abdollahi, S. Akbari and H. R. Maimani in (Non-commuting graph of a group, Journal of Algebra, 298 (2006) 468-492.)

Order Conjecture. If G and H are two non-abelian finite groupsMathworldPlanetmath with isomorphicPlanetmathPlanetmathPlanetmathPlanetmath non-commuting graphs, then |G|=|H|.

It was proved that Order Conjecture is true if and only if it is true for all non-abelian solvablePlanetmathPlanetmath finite AC-groups. By an AC-group, we mean a group in which the centralizerMathworldPlanetmathPlanetmathPlanetmath of every non-central element is abelianMathworldPlanetmath.

The order Conjecture has been refuted in the following paper

[*] A. R. Moghaddamfar, On non-commutating graphs, Siberian Math. J. 47 (2006), no. 5, 911-914.

It is mentioned in [*] that the example given in the article is due to M. Isaacs.

Title Order Conjecture for non-commuting graph of a group
Canonical name OrderConjectureForNoncommutingGraphOfAGroup
Date of creation 2013-03-22 15:18:53
Last modified on 2013-03-22 15:18:53
Owner abdollahi (9611)
Last modified by abdollahi (9611)
Numerical id 10
Author abdollahi (9611)
Entry type Conjecture
Classification msc 20D60