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.

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
.
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