# On the commutativity degree of metacyclic groups of 5-power order using conjugation action

## DOI:

https://doi.org/10.11113/mjfas.v0n0.695## Keywords:

Commutativity degree, metacyclic 5-groups, conjugation action## Abstract

An extension of the concept of commutativity degree named the probability that an element of a group fixes a set was introduced in 2013. Suppose is a metacyclic 5-group and is the set of all ordered pairs (x,y) in G*G such that lcm(|x|,|y|)=5, xy=yx and x is not equal to y. In this paper, the probability that an element of a metacyclic 5-group fixes the set is computed by using conjugation action.

## References

Erdos, P. and Turan, P. 1965. On some problems of a statistical group theory. Probability Theory and Related Fields. 4(2), 175-186.

Gustafson, W. 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 commutator of two group elements is equal to a given element. Journal of Pure and Applied Algebra. 212, 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.

Gastekz, A. and Kuzmanovich, J. 2010. Commutativity Degree of a Finite Group. Ms.C. Wake Forest University, North Carolina.

Barzgar, R., Erfanian, A. and Farrokhi, M. 2016. Probability of mutually commuting two subsets of a finite group. Ars Combinatoria. CXXIV, 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 set by conjugation. International Journal of Applied Mathematics and Statistics. 32(2), 111-117.

El-Sanfaz, M. A. 2016. The probability that an element of a non-abelian group fixes a set and its applications in graph theory, Doctor of Philosophy Thesis, Universiti Teknologi Malaysia.

Rose, J. S. 1978. A course on group theory. Great Britain, Cambridge: Cambridge University Press.

Rotman, J. J. 2002. Advanced modern algebra. Upper Saddle River, New Jersey: Prentice Hall.

Goodman, F. 2003. Algebra: abstract and concrete. (2nd Edition). Upper Saddle River, New Jersey: Prentice Hall.

Dummit, D. S. and Foote, R. M. 2004. Abstract algebra. (3rd Edition). United States of America: John Wiley & Sons, Inc.

Basri, A. M. A. 2013. Capability and homological functors of finite metacyclic p-groups. Doctor of Philosophy Thesis. Universiti Teknologi Malaysia.

Sherman, G. 1975. What is the probability an automorphism fixes a group element? The American Mathematical Monthly. 82(3), 261-264.

Moghaddam, M. R. R. Saeedi, F. and Khamseh, E. 2011. The probability of an automorphism fixing a subgroup element of a finite group. Asian-European Journal of Mathematics. 4(2), 301-308.