# 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 groups^{} with isomorphic^{} 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 solvable^{} finite $AC$-groups.
By an $AC$-group, we mean a group in which the centralizer^{} of every non-central element is abelian^{}.

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 |
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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 |