The graph of equivalence classes and Isoclinism of groups

Rezaei, R.; Varmazyar, M.

The graph of equivalence classes and Isoclinism of groups

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎Malayer University‎, ‎Malayer‎, ‎Iran.

Abstract

‎Let $G$ be a non-abelian group and let $\Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $\sim$ on the set of $V(\Gamma(G))=G\setminus Z(G)$ by taking $x\sim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)=\{u\in G \ | \ x \textrm{ and } u \textrm{ are adjacent in }\Gamma(G)\}$ is the open neighborhood of $x$ in $\Gamma(G)$‎. ‎We introduce a new graph determined by equivalence classes of non-central elements of $G$‎, ‎denoted $\Gamma_E(G)$‎, ‎as the graph whose vertices are $\{[x] \ | \ x \in G\setminus Z(G)\}$ and join two distinct vertices $[x]$ and $[y]$‎, ‎whenever $[x,y]\neq 1$‎. ‎We prove that group $G$ is AC-group if and only if $\Gamma_E(G)$ is complete graph‎. ‎Among other results‎, ‎we show that the graphs of equivalence classes of non-commuting graph associated with two isoclinic groups are isomorphic.

Keywords

Main Subjects

05-XX Combinatorics

References

A. Abdollahi, Engel graph associated with a group, J. Algebra 318 (2007) 680--691.

A. Abdollahi, S. Akbari and H.R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006) 468--492.

A. Abdollahi and A. Mohammadi Hassanabadi, Finite groups with a certain number of elements pairwise generating a non-nilpotent subgroup, Bull. Iranian Math. Soc. 30 (2004), no. 2, 1--20.

A. Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Comm. Algebra 35 (2007) 2057--2081.

S.M. Belcastro and G.J. Sherman, Counting centralizers in finite groups, Math. Mag 67 (1994), no. 5, 366--374.

E.A. Bertram, M. Herzog and A. Mann, On a graph related to conjugacy classes of groups, Bull. Lond. Math. Soc. 22 (1990), no. 6, 569--575.

A. M. Broshi, Finite groups whose Sylow subgroups are abelian, J. Algebra 17 (1971) 74--82.

The GAP Group, GAP Groups, Algorithms, and Programming, version 4.4.10, 2007.

P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940) 130--141.

P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995) 847--869.

A.R. Moghaddamfar, W.J. Shi, W. Zhou and A.R. Zokai, On noncommuting graphs associated with a finite group, Sib. Math. J. 46 (2005), no. 2, 325--332.

B.H. Neumann, A problem of Paul Erdos on groups, J. Aust. Math. Soc. Ser. A 21 (1976) 467--472.

L. Pyber, Non-commuting elements and the index of the centre in a finite group, J. Lond. Math. Soc. (2) 35 (1987), no. 2, 287--295.

R. Rezaei and A. Erfanian, A note on the relative commutativity degree of finite groups, Asian-Eur. J. Math. 7 (2014), no. 1, 1--9.