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.