Fuzzy subgroups of the direct product of a generalized quaternion group and a cyclic group of any odd order

Bentea and T\u{a}rn\u{a}uceanu~(An. \c{S}tiin\c{t}. Univ. Al. I.Cuza Ia\c{s}, Ser. Nou\v{a}, Mat., {\bf ۵۴(۱)} (۲۰۰۸), ۲۰۹-۲۲۰)proposed the following problem: Find an explicit formula for thenumber of fuzzy subgroups of a finite hamiltonian group of typeQ_۸\times \mathbb{Z}_n where Q_۸ is the quaternion group oforder ۸ and n is an arbitrary odd integer. In this paper weconsider more general group: the direct product of a generalizedquaternion group of any even order and a cyclic group of any oddorder. For this group we give an explicit formula for the number offuzzy subgroups.

کلیدواژه ها:

Generalized quaternion group ، Hamiltonian group ، Fuzzy subgroups ، Subgroup chain ، Generating function