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

Document Type: Research Paper

Author

Department of Mathematics, Gangneung-Wonju National University, 7, Jukheon-gil, Gangneung-si, Gangwon-do 210-702, Republic of Korea

Abstract

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 54(1)} (2008), 209-220)
proposed the following problem: Find an explicit formula for the
number of fuzzy subgroups of a finite hamiltonian group of type
$Q_8\times \mathbb{Z}_n$ where $Q_8$ is the quaternion group of
order $8$ and $n$ is an arbitrary odd integer. In this paper we
consider more general group: the direct product of a generalized
quaternion group of any even order and a cyclic group of any odd
order. For this group we give an explicit formula for the number of
fuzzy subgroups.

Keywords

### References

bibitem{AI79} M. Aigner, {it Combinatorial theory}, Springer-Verlag, New York Inc., 1979.

bibitem{BT08} L. Bentea and M. Tu{a}rnu{a}uceanu, {em A note on the number of fuzzy subgroups of finite groups},
An. c{S}tiinc{t}. Univ. Al. I. Cuza Iac{s}, Ser. Nouv{a}, Mat.,
{bf 54(1)} (2008), 209-220.

bibitem{DA81} P. S. Das, {em Fuzzy groups and level subgroups},
J. Math. Anal. Appl., {bf  84} (1981), 264-269.

bibitem{MM01} V. Murali and B. B. Makamba, {em On an equivalence of fuzzy subgroups I},
Fuzzy Sets and Systems, {bf 123} (2001), 259-264.

bibitem{MM03} V. Murali and B. B. Makamba, {em On an equivalence of fuzzy subgroups II},
Fuzzy Sets and Systems, {bf 136} (2003), 93-104.

bibitem{MM04} V. Murali and B. B. Makamba, {em Counting the number of fuzzy subgroups of an abelian group of
order $p^nq^m$}, Fuzzy Sets and Systems, {bf 144} (2004), 459-470.

bibitem{MMV04} V. Murali and B. B. Makamba, {em Fuzzy subgroups of finite abelian groups},
FJMS, {bf 14} (2004), 113-125.

bibitem{JJpre} J. M. Oh, {em The number of chains of subgroups of a finite cyclic group},
European J. Combin., {bf 33} (2012), 259-266.

bibitem{RO94} J. S. Rose, {it A course on group theory}, Dover publications, Inc., New York, 1994.

bibitem{SC64} W. R. Scott, {it Group theory}, Prentice-Hall, Englewood Cliffs, NJ, 1964.

bibitem{TB08} M. Tu{a}rnu{a}uceanu and L. Bentea, {em On the number of fuzzy subgroups of finite abelian
groups}, Fuzzy Sets and Systems, {bf 159} (2008), 1084-1096.

bibitem{Tu95} A. Tucker, {it Applied combinatorics}, John Wiley & Sons, Inc., New York, 1995.

bibitem{WO04} A. C. Volf, {em Conuting fuzzy subgroups and chains of subgroups},
Fuzzy Systems & Artificial Intelligence, {bf 10(3)} (2004),
191-200.