Khaled Alajmi
Department of Mathematics, Public Authority for Applied Education and Training, Ardiyah, Kuwait.
*Corresponding author:Khaled Alajmi
Abstract
Let G be a finite group. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute E(G) for certain nilpotent groups. This paper is divided into six sections. The first section is an introduction. In the second section we present the definition λn (G) the probability that n elements drawn at random, with replacement from a finite group G generate G, and the definition E(G) the expectation of the group G, that is the expected number of elements of G which have to be drawn at random, with replacement, before a set of generators is found. In the third section we present some preliminaries and earlier results e.g if G is a group and Φ(G) is its Frattini subgroup. Then, λn (G)=λn (G/Φ(G)). In section four, we computed E(G) for certain nilpotent groups such as
and
In section five, we computed E(G) for a non-trivial example. The last section is a conclusion.
References
[1] Acciaro, V. (1996). The probability of generating some common families of finite groups. Utilitas Mathematica, 49, 243-253.
[2] Alhasanat, B. (2022). On classification of groups of order p^4, p odd prime, International Journal of Mathematics and Computer Science, 17 no 4, 1564-1593.
[3] Alhasanat, B. (2022). Some probabilistic approaches on finite groups, International Journal of Applied Mathematics, 35 NO.6, 827-838.
[4] Atkinson, M. (1990). A survey of algorithms for handling permutation group. School of Computer Science Technical report SCS-TR-164, Carlton University, Ottawa, January.
[5] Eberhard, S, Shumyatsky, P. (2023). Probabilistically nilpotent groups of class 2. Mathematische Annalen,
doi.org 10.1007/500208-023-02567-0- accepted and to appear.
[6] Hall, P. (1936). The Eulerian functions of a group. Quart. J.Math., Ox. Series, 7, 134-151.
[7] Madadi, H. (2023). An upper bound for the probability of generating a finite nilpotent group, Kyungpook Math. J., 63, 167-173.
How to cite this paper
On Expectation of Generating Certain Finite Nilpotent Groups
How to cite this paper: Khaled Alajmi. (2024) On Expectation of Generating Certain Finite Nilpotent Groups. Journal of Applied Mathematics and Computation, 8(2), 113-119.
DOI: http://dx.doi.org/10.26855/jamc.2024.06.003