Commuting Conjugacy Class Graph of G when G / Z(G)~=D2n

Document Type : Original Scientific Paper


Department of Mathematics, Savadkooh Branch, Islamic Azad University, Savadkooh, Iran.


Suppose G is a finite non-abelian group and Γ(G) is a simple graph with the non-central conjugacy classes of G as its vertex set. Two different noncentral conjugacy classes C and B are assumed to be adjacent in Γ(G) if and only if there are elements a ∈ A and b ∈ B such that ab = ba. This graph is called the commuting conjugacy class graph of G. In this paper, the structure of the commuting conjugacy class graph of a group G with this property that Z(G)/G≅D_{2n} will be determined.


[1] A. Abdollahi, S. M. Jafarian Amiri and A. Mohammadi Hassanabadi, Groups with specific number of centralizers, Houston J. Math. 33 (1) (2007) 43 - 57.
[2] S. J. Baishya, On finite groups with specific number of centralizers,
Int. Electron. J. Alg. 13 (2013) 53 - 62.
[3] S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, 
Math. Mag. 5 (1994) 111 - 114.
[4] F. Harary,
Graph Theory, Addison-Wesley, Reading, MA, 1969.
[5] A. Mohammadian, A. Erfanian, M. Farrokhi D. G. and B. Wilkens, Trianglefree commuting conjugacy class graphs,
J. Group Theory 19 (3) (2016) 1049-1061.
[6] D. J. Robinson,
A Course in the Theory of Groups, Springer-Verlag, New York, 1996.
[7] M. A. Salahshour and A. R. Ashrafi, Commuting conjugacy class graph of finite CA-groups,
Khayyam J. Math. 6 (1) (2020) 108–118.
[8] R. Schmidt, Zentralisatorverbände endlicher Gruppen (German),
Rend. Sem. Mat. Univ. Padova 44 (1970) 97 - 131.
[9] The GAP Team, Group, GAP - Groups, Algorithms, and Programming, Version 4.5.5, 2012,