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THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP
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 Title & Authors
THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP
Mirzargar, Mahsa; Pach, Peter P.; Ashrafi, A.R.;
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 Abstract
Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and () joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by . The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut() is abelian if and only if ; is of prime power if and only if , and is square-free if and only if . Some new graphs that are useful in studying the automorphism group of are presented and their main properties are investigated.
 Keywords
commuting graph;automorphism group;extra special group;
 Language
English
 Cited by
 References
1.
A. Abdollahi, S. Akbari, and H. R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006), no. 2, 468-492. crossref(new window)

2.
N. Biggs, Algebraic Graph Theory, Second ed., Cambridge Univ. Press, Cambridge, 1993.

3.
P. J. Cameron, Automorphisms of graphs, Beineke, Lowell W. (ed.) et al., Topics in Algebraic Graph Theory, Cambridge: Cambridge University Press, Encyclopedia of Mathematics and Its Applications 102 (2004), 137-155.

4.
R. Frucht, On the groups of repeated graphs, Bull. Amer. Math. Soc. 55 (1949), 418-420. crossref(new window)

5.
The GAP Team, GAP, Groups, Algorithms and Programming, Lehrstuhl De fur Mathematik, RWTH, Aachen, 1995.

6.
D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Mathematical Surveys and Monographs, 40.1. American Mathematical Society, Providence, RI, 1994.

7.
J. Lennox and J. Wiegold, Extension of a problem of Paul Erdos on groups, J. Aust. Math. Soc. Ser. A 31 (1981), no. 4, 459-463. crossref(new window)

8.
M. Mirzargar and A. R. Ashrafi, Some distance-based topological indices of a noncommuting graph, Hacet. J. Math. Stat. 41 (2012), no. 4, 515-526.

9.
A. R. Moghaddamfar, On noncommutativity graphs, Siberian Math. J. 47 (2006), 911-914. crossref(new window)

10.
A. R. Moghaddamfar, W. J. Shi, W. Zhou, and A. R. Zokayi, On the noncommuting graph associated with a finite group, Siberian Math. J. 46 (2005), no. 2, 325-332. crossref(new window)

11.
B. H. Neumann, A problem of Paul Erdos on groups, J. Aust. Math. Soc. Ser. A 21 (1976), no. 4, 467-472. crossref(new window)

12.
D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.

13.
D. M. Rocke, p-groups with abelian centralizers, Proc. London Math. Soc. (3) 30 (1975), 55-75.

14.
D. B. West, Introduction to Graph Theory, Prentice Hall. Inc. Upper Saddle River, NJ, 1996.

15.
D. L. Winter, The automorphism group of an extraspecial p-group, Rocky Mountain J. Math. 2 (1972), no. 2, 159-168. crossref(new window)