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

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DOI: http://dx.doi.org/10.19184/ijc.2024.8.1.3

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