In this paper, let G be a metacyclic 3-group of negative type of nilpotency class at least three. Let Ω be the set of all subsets of commuting elements of G of size three in the form of (a,b), where a and b commute and lcm(|a|, |b|) = 3. The probability that an element of a group G fixes a set Ω is considered as one of the extensions of the commutativity degree that can be obtained under group actions on a set. In this paper, we compute the probability that an element of G fixes a set Ω in which G acts on Ω by conjugation. The results are then applied to graph theory, more precisely to orbit graph.
REFERENCES
1.
G. A.
Milller
, “Relative number of a non-invariant operators in a group
”, in Proceedings of the National Academy of Sciences
, 1994
, 30
(2
), pp. 25
–28
.2.
P.
Erdos
and P.
Turan
, Acta Math. Acad. Of Sci. Hung
19
, 413
–435
(1968
).3.
W. H.
Gustafson
, Amer. Math. Monthly
80
, 1031
–1034
(1973
).4.
D.
MacHale
, The Mathematical Gazette
58
, 199
–202
(1974
).5.
S. M. S.
Omer
, N. H.
Sarmin
, A.
Erfanian
and K.
Moradipour
, International Journal of Applied Mathematics and Statistics
32
(2
), 111
–117
(2013
).6.
C.
Reis
, Abstract Algebra: An Introduction to Groups, Rings and Fields
(World Scientific
, Hackensack, New Jersey
, 2011
).7.
W. K.
Nicholson
, 4th Ed. Introduction to Abstract Algebra
(John Wiley & Sons, Inc.
, Hoboken, New Jersey
, 2012
).8.
J. J.
Rotman
, Advanced Modern Algebra
(Prentice Hall
, Upper Saddle River, New Jersey
, 2002
).9.
J. R.
Beuerle
, Algebra Colloquium
12
(4
), 553
–562
(2005
).10.
F. M.
Goodman
, 2nd Ed. Algebra: abstract and concrete: stressing symmetry
(Prentice Hall/ Pearson Education
, Upper Saddle River, New Jersey
, 2003
).11.
J.
Bondy
and G.
Murty
, 5th Ed. Graph Theory with Application
(North Holland
, Boston New York
, 1982
).12.
C.
Godsil
and G.
Royle
, 5th Ed. Algebraic Graph Theory
(Springer
, Boston New York
, 2001
).13.
G. J.
Sherman
, Amer. Math. Monthly
82
(3
), 261
–264
(1975
).14.
M. R. R.
Moghaddam
, F.
Saeedi
and E.
Khamseh
, Asian-European Journal of Mathematics
4
(2
), 301
–308
(2011
).15.
S. M. S.
Omer
, N. H.
Sarmin
and A.
Erfanian
, Applied Sciences Journal
27
(12
), 1637
–1642
(2013
).16.
M. A.
El-sanfaz
, N. H.
Sarmin
and S. M. S.
Omer
, International Journal of Applied Mathematics and Statistics
52
(1
), 1
–6
(2014
).17.
M.
Bianchi
, D.
Chillag
, A.
Mauri
, A.
Herzog
and C.
Scoppola
, Arch Math
58
, 126
–132
(1992
).18.
A.
Moreto
, G.
Qian
and W.
Shi
, Arch Math
85
, 101
–107
(2005
).19.
K.
Moradipour
, N.H.
Sarmin
, and A.
Erfanian
, Journal of Basic and Applied Scientific Research
, 3
, 898
–902
(2013
).20.
A.
Erfanian
and B.
Tolue
, Conjugate Graphs of Finite Groups, Discrete Mathematics, Algorithms and Applications
4
, 35
–43
(2012
).21.
S. M. S.
Omer
, “Extension of the commutativity degree of some finite groups and their related graphs
,” Ph.D. thesis, Universiti Teknologi
Malaysia
, 2014
.22.
S.M.S.
Omer
, N.H.
Sarmin
, and A.
Erfanian
, “The Orbit Graph for Some Finite Solvable Groups
”, in The Proceedings of The 3rd International Conference on Mathematical Sciences
, AIP Conference Proceedings
1602
, 2014
, pp. 863
.23.
S.M.S.
Omer
, N.H.
Sarmin
, and A.
Erfanian
, World Applied Sciences Journal
27
, 1637
–1642
(2013
).24.
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
.
This content is only available via PDF.
© 2016 Author(s).
2016
Author(s)