On the generating graph of a finite group

Haval M. Mohammed Salih

Abstract


In this paper, we study the generating graph for some finite groups which are semi-direct product ℤn ⋊ ℤm (direct product ℤn × ℤm) of cyclic groups ℤn and ℤm. We show that the generating graphs of them are regular (bi-regular, tri-regular) connected graph with diameter 2 and girth 3 if n and m are prime numbers. Several graph properties are obtained. Furthermore, the probability that 2-randomly elements that generate a finite group G is P(G) = |{(a,b) ∈ G×G|G=❬a,b❭}|/|G|2. We find the general formula for P(G) of given groups. Our computations are done with the aid of GAP and the YAGs package.

Keywords


Semi-Direct Product Group, Generating Graph and Probability

Full Text:

PDF

References

E. Bertram, M. Herzog, and A. Mann, On a graph related to conjugacy classes of groups, Bulletin of the London Mathematical Society, 22(6), (1990), 569–575.

T. Breuer, R. Guralnick, and W. Kantor, Probabilistic generation of finite simple groups, II Journal of Algebra, 320(2), (2008), 443–494.

A. Erfanian and B. Tolue, Conjugate graphs of finite groups, Discrete Mathematics, Algorithms and Applications, 4(2) (2012), 1250035.

B. Esther, Probability of generating a dicyclic group using two elements Pi Mu Epsilon Journal, 14(3), (2015), 165–168.

M. W. Liebeck and A. Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, J. Algebra, 184 (1996), 31–57.

K. L. Patti, The probability of randomly generating a finite group Pi Mu Epsilon Journal, 11 (6), (2002), 313–316.

H. Tong-Viet, Finite groups whose prime graphs are regular, Journal of Algebra, 397 (2014), 18–31.

R. J. Wilson, Introduction to graph theory, Pearson Education India, (1979).


Refbacks

  • There are currently no refbacks.


ISSN: 2541-2205

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View IJC Stats