In this paper, G denotes a dihedral group of order 2n and Ω denotes the set of all subsets of all commuting elements of size two in the form of (a,b), where a and b commute and |a| = |b| = 2. By extending the concept of commutativity degree, the probability that an element of a group fixes a set can be acquired using the group actions on set. In this paper, the probability that an element of G fixes the set Ω under regular action is computed. The results obtained are then applied to graph theory, more precisely to generalized conjugacy class graph and orbit graph.
REFERENCES
1.
G.
Miller
, Proc. Nat. Acad. Sci. USA
30
(2
, 25
–28
(1944
).2.
G.
Gustafson
, The American Mathematical Monthly
80
(9
, 1031
–1034
(1973
).3.
D.
MacHale
, The Mathematical Gazette
58
(405
, 199
–202
(1974
).4.
M. A.
El-sanfaz
, N. H.
Sarmin
and S. M. S.
Omer
, International Journal of Applied Mathematics and Statistics
52
(1
, 1
–6
(2014
).5.
J.
Bondy
and G.
Murty
, 5th Ed. Graph Theory with Application
(North Holand
, Boston New York
, 1982
).6.
C.
Godsil
and G.
Royle
, 4th Ed. Algebraic Graph Theory
(Springer
, Boston New York
, 2001
).7.
J. K.
Xu
, 5th Ed. Theory and Application of Graph
(Academic Publishers
, Boston New York
, 2003
).8.
E. A.
Bertram
, M.
Herzog
and A.
Man
, Bull London Math Soc.
, 22
, 569
–575
(1990
).9.
S. M. S.
Omer
, N. H.
Sarmin
and A.
Erfanian
, “Generalized conjugacy class graph of some finite non-abelian groups
,” in International Conference on Mathematics, Engineering and Industrial Applications 2014
, AIP Conference Proceedings
1660
, (2015
), pp. 010001
.10.
S. M. S.
Omer
, N. H.
Sarmin
and A.
Erfanian
, International Journal of Pure and Applied Mathematics
102
(4
, 747
–755
(2015
).11.
M. A.
El-sanfaz
, N. H.
Sarmin
and S. M. S.
Omer
, World Applied Sciences Journal
32
(3
, 459
–464
(2014
).12.
M. A.
El-sanfaz
, N. H.
Sarmin
and S. M. S.
Omer
, Jurnal Teknologi
71
(1
, 7
–10
(2014
).13.
M. A.
El-sanfaz
N. H.
Sarmin
and S. M. S.
Omer
, International Journal of Mathematical Analysis
. 9
(4
), 161
–167
(2014
).14.
M. A.
El-sanfaz
and N. H.
Sarmin
, Global Journal of Pure and Applied Mathematics
11
(2
, 899
–908
(2015
).15.
M.
Bianchi
, D.
Chillag
, A.
Mauri
, M.
Herzog
and C.
Scoppola
, Arch Math
58
, 126
–132
(1992
).16.
A.
Moreto
, G.
Qian
and W.
Shi
, Arch. Math
85
, 101
–107
(2005
).17.
18.
K.
Moradipour
, N. H.
Sarmin
and A.
Erfanian
, Journal of Basic and Applied Scientific Research
. 3
(1
, 898
–902
(2013
).19.
N. H.
Sarmin
, A.
Erfanian
and S. M. S.
Omer
, Some applications of metacyclic 2-groups of negative type in
The 3rd International Conference on Computer Engineering and Mathematical Sciences, Conference proceedings
, (2014
), pp. 120
–124
.20.
S. M. S.
Omer
, N. H.
Sarmin
and A.
Erfanian
, in The Proceedings of The 3rd International Conference on Mathematical Sciences
, AIP Conference Proceedings
1602
, (2014
), pp. 863
.21.
S. M. S.
Omer
, N. H.
Sarmin
and A.
Erfanian
, Indian Journal of Science and Technology
7
(12
, 2113
–2117
(2014
).22.
This content is only available via PDF.
© 2016 Author(s).
2016
Author(s)
