Microsoft Word - 40.doc Mathematical Problems of Computer Science 38, 93--94, 2012. 93 About Order 3 Hypergroups Over Group Which Arisе From Groups of Order 18 Pervaneh Zolfaghari Yerevan State University Each (unitary) hypergroup M over a group H is isomorphic to a hypergroup over group associated with a complementary set M to a subgroup H of a group G. All unitary hypergroups of order 3 over group, associated with the complementary sets to a subgroup of symmetric groups S3 and S4 was described in [1] and [2]. According to [3] any hypergroup over group is reduced to an irreducible hypergroup over group. Therefore to describe all hypergroups over group first of all it is necessary to find all irreducible hypergroups over group. Let M be a hypergroup over a group H. Then there exists a structural homomorphism from H to the SM. The hypergroup over group is irreducible if and only if the kernel of this homomorphism is trivial. Suppose |M| = 3. Then we have the following sorts of monomorphisms from H to SM. 1) Trivial |H| = 1. In this case the hypergroup over group is a group. 2) |H| = 2. In this case the hypergroup M is isomorphic to a hypergroup associated with a subgroup of index 3 in S3. There exist three such hypergroups over group (up to isomorphism). One of this hypergroups is reducible and is reduced to a group. Two others are irreducible. 3) |H| = 3. Then the hypergroup over group is reduced to a group, because there does not exist a non-abelian group of order 9. 4) |H| = 6. Every such hypergroup over group is associated with a complementary subset M to a subgroup H of index 3 in a group G of order 18. Up to isomorphism, there exist three non-abelian groups G of order 18. (a) G =  a, b, c, a3 = b2 = c3 = e, ba = a2b, ca = ac, cb = bc  . This group is the direct product of its symmetric subgroup N =  a, b  S3, and cyclic subgroup  c  C3. It has four subgroups of index 3: one normal subgroup N and three conjugate cyclic subgroups, which are generated by c and one of elements of order 2 in N. (b) G =  a, b, a9 = b2 = baba = e  . Then G = D9 is the dihedral group and is the semidirect product of the normal subgroup N =  a   C9, and cyclic subgroup  b   C2. This group has three subgroups of order 6. They are conjugate and isomorphic to S3. (c) G =  a, b, c, a3 = b3 = c2 = e, ba = ab, ca = ac, cb = bc  . 94 About Order 3 Hypergroups Over Group Which Arisе From Groups of Order 18 This group is a semidirect product of the normal subgroup N =  a, b  C3 C3 and cyclic subgroup  c  C2. It has 12 subgroups of index 3. They are generated by one element of order 3 and one element of order 2. Two subgroups of order 6 are conjugate if and only if they have the same generating element of order 3. Thus, there are four classes consisting of three conjugate subgroups of order 6 In case (a) for an arbitrary subgroup H of order 6 of the group G the structural homomorphism from H to SM has a nontrivial kernel, i. e. in this case we have not irreducible hypergroups over group. The description of order 3 hypergroups over group, arising in the cases (b) and (c), is more complicate. Reference 1. Zolfaghary P., The hypergroups of order 3, arising from symmetric group S3. Fourth Group Theory Conference of Iran, Payam Noor University of Isfahan, Isfahan, Iran, March 7-9, 2012 . 2. Zolfaghary P., The order three right hypergroups over group, arising from symmetric group S4. Thes. of Conf. of AMU, Yerevan, 25 may -2 Junе 2012. p. 94-96. 3. Dalalyan S. H., The reducibility theory for hypergroups over group. Thes. of Conf. of AMU, Yerevan, 25 may -2 Juin 2012. p. 22 - 24.