On finite groups of whose all proper subgroups are

Hossein AndikfarAli Reza Ashrafi



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A finite group G is called w-cyclic, if G has at most d subgroup, for all divisors d of |G|. In this paper, we study the structure of a finite group all of whose proper subgroups are w-cyclic. In the case that G has prime power order, we prove that such a group is elementary abelian of order p2, p is prime, the quaternion group Q8 or the generalized quaternion group Q16. We prove that if such a G is not a p-group, then G is solvable and in some cases, we obtain the structure of G. Finally, we characterize the finite groups with w-cyclic proper quotient groups.

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Andikfar, Hossein, Ashrafi, Ali Reza