The Conjecture of Group Structure: The Relationship Between The Alpha Invariant and Nilpotency in Finite Groups

Abstract

In this research work, we acknowledge and explore the relation between the alpha value and non-nilpotent groups, leading to the proof of a conjecture put forward in research by Cayley (2021). We demonstrate that if 𝐺 is non-nilpotent and 𝛼(𝐺) = 􀬷 􀬸 then 𝐺 ≅ 𝐷􀬶􀬸 × 𝐶􀬶􀳙 , with a nontrivial centre, where 𝑛 ∈ {0, 1}. Furthermore, we conclude that the conjecture holds for 𝐺 ≅ 𝐷􀬶􀬸 × 𝐶􀬶􀳙 as well. We again prove, using both computational and theoretical techniques, that a subgroup which is nontrivial in 𝐺 exists with both normal and characteristic properties. We finally prove a theorem related to the count involving subgroups, cyclic in nature, of finite groups 𝐺 where |𝐶(𝐺)| = |𝐺| − 6. Thus, we demonstrate that if 𝐺 is one of the groups 𝐷􀬶􀬸, 𝐶􀬵􀬶, 𝐶􀬽, 𝐶􀬵􀬴, 𝐷􀬵􀬼, or 𝐷􀬶􀬴, then |𝐶(𝐺)| = |𝐺| − 6.