The Conjugation Degree on a Set of Metacyclic 3-Groups
DOI:
https://doi.org/10.11113/mjfas.v16n5.1925Keywords:
Commutativity degree, Conjugation action, Conjugation degree, Metacyclic groupsAbstract
Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements x and y in a group G commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ω is explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to depend on the order of Ω.
References
Erd¨os, P. and Tur´an, P. 1965. On some problems of a statistical group theory i. Probability Theory and Related Fields. 4(2), 175-186.
Erd¨os, P. and Tur´an, P. 1968. On some problems of a statistical group theory iv. Acta Mathematica Academiae Scientarium Hungaricae Tomus, 19(3-4): 413-435. Erd¨os, P. and Tur´an, P. 1967a. On some problems of a statistical group theory ii. Acta Mathematica Academiae Scientarium Hungaricae Tomus, 18(1-2): 151.
Erd¨os, P. and Tur´an, P. 1967. On some problems of a statiscal group theory iii. Acta Mathematica Academiae Scientarium Hungaricae Tomus, 18(3-4): 309-320.
Gustafson, W.H. 1973. What is the probability that two group elements commute? The American Mathematical Monthly. 80(9), 1031-1034.
Pournaki, M.R. and Sobhani, R. 2008. Probability that the commutator of two group elements is equal to a given element. Journal of Pure and Applied Algebra. 212(4), 727-734.
Alghamdi, A.M. and Russo, F.G. 2012. A generalized of the probability that commutator of two group elements is equal to a given element. Bulletin of Iranian Mathematical Society. 38(4), 973-986.
Erfanian, A., Rezaei, A. and Lescot, P. 2007. On the relative commutativity degree of a subgroup of a finite group. Communications in Algebra. 35(12), 4183-4197.
Castelaz, A. 2010. Commutativity degree of a finite group. Masters thesis, Wake Forest University, North Calorina.
Barzgar, R.P., Erfanian, A. and Farrokhi, M.D.G. 2016. Probability of mutually commuting two finite subsets of a finite group. Ars Combinatoria. 124, 165-176.
Omer, S.M.S., Sarmin, N.H., Erfanian, A. and Moradipour, K. 2013. The probability that an element of a group fixes a set and the group act on a aset by conjugation. International Journal of Applied Mathematics and Statistics. 32(2), 111-117. Rose, J.S. 1994. A course on group theory. Courier Corporation.
Rotman, J.J. 2010. Advanced modern algebra, volume 114. American Mathematical Society.
Basri, A.M.A. 2014. Capability and Homological Functors of Finite Metacyclic p-Groups. PhD thesis. Universiti Teknologi Malaysia.
Goodman, F. 2006. Algebra, Abstract and Concrete. SemiSimple Press. 2006.
Sherman, G. 1975. What is the probability an automorphism fixes a group element? The American Mathematical Monthly. 82(3), 261-264.
El-Sanfaz, M.A. 2016. The Probability that an Element of a Non-Abelian Group Fixes a Set and Its Applications in Graph Theory. PhD thesis, Universiti Teknologi Malaysia.
