CHARACTER TABLE GROUPS AND EXTRACTED SIMPLE AND CYCLIC POLYGROUPS

Let G be a finite group and Ĝ be the set of all irreducible complex characters of G. In this paper, we consider Ĝ as a polygroup. We call that Ĝ simple if it has no proper normal subpolygroup and show that if Ĝ is a single power cyclic polygroup, then Ĝ is a simple polygroup and hence the polygroups wich are indeuced by symmetric group and alternating group are simple. Also, we prove that if G is a non-abelian simple group, then Ĝ is a single power cyclic polygroup. Moreover, we classify D̂2n for all n. Also, we investigate the same property for two other groups.


INTRODUCTION
Let G be a finite group and Irr(G) = {χ 1 , χ 2 , ..., χ k } be the set of all irreducible characters of G. Brauer [1] introduced the idea of studying character tables considering them as square arrays of complex numbers satisfying certain conditions. Afterwards, Comer [2] described several hypergroup constructions based on assumptions which arise in the study of symmetry. In particular, he showed that a natural hypergroup is associated with every character algebra. Hence the hypergroup suggests itself as a tool for group theorists. These concepts provide a new language in which groups and their character tables can be fruitfully discussed.
Roth in [11] studied canonical hypergroups <Ĝ, * , χ 1 , − >, whereĜ = Irr(G), χ 1 is the trivial character and the product χ i * χ j is the set of those irreducible constituents which appear in the element wise product χ i χ j . Furthermore,χ, the complex conjugate of χ, is the inverse of χ. Canonical hypergroups were introduced by Mittas [10] and later, Corsini in [3] analyses a particular type of canonical hypergroups and their homomorphisms. Afterwards, Comer introduced this class of hypergroups independently, using the name of polygroups and pointed out that polygroups have application in color schemes and combinatorics [2]. We call the polygroupĜ the "character polygroup". The structure of polygroups is similar to groups, since identity and inverse elements exist in polygroups. So, we say P =< P, ·, e, −1 >, is a polygroup if P is a non-empty set, e ∈ P, −1 is a unitary operation on P, · maps P × P into the set of all non-empty subsets of P, and the following axioms hold for all x, y, z ∈ P : (1) (x · y) · z = x · (y · z); (2) e · x = x · e = x; (3) x ∈ y · z implies that y ∈ x · z −1 and z ∈ y −1 · x.
A non-empty subset K of a polygroup P is a subpolygroup of P if (1) a, b ∈ K implies that a · b ⊆ K, (2) a ∈ K implies that a −1 ∈ K.
Since polygroups have properties close to groups, concepts such as normal subgroups and isomorphism theorems are defined for them . A subpolygroup N of a polygroup P is normal in P if a −1 N a ⊆ N, for all a ∈ P see [4]. We say that the polygroup P is simple if it does not have any proper normal subpolygroups. So if G is a simple group, thenĜ is a simple polygroup. It is clear thatĜ is always a normal subpolygroup and the trivial subpolygroup {χ 1 } is not a normal subpolygroup ofĜ. The Classification of simple groups has been of interest to mathematicians for many years.
Roth in [11], showed that the mapping K −→ G K yields a one to one correspondence between the set of all normal subgroups of G and the set of subpolygroups ofĜ. In this paper, we investigate the simplicity of a character polygroup G. In fact, we show that ifĜ is a single power cyclic polygroup, then it is a simple polygroup. The proof of this result is carried out using the fundamental relations on hypergroups. Cyclic hypergroups are a certain subclass of hypergroups. Cyclic hypergroups were first initiated by Wall [15] and afterwards have been studied by Vougiouklis [13], Konguetsof et al. [6] and Leoreanu [8]. The hypergroup (H, •) is called cyclic with finite period respect to h ∈ H if there exists a positive integer s ∈ Z + , such that The minimum of all such positive integers s is called the period of the generator h. If there exists k ∈ Z + , such that then H is called a single power cyclic hypergroup and h is a generator of H. The minimum of all such positive integer k is called the period of the generator h.
In [12], we showed thatŜ n for n ≥ 3 andÂ n for n ≥ 4 are single power cyclic polygroups. An obvious conclusion of this result is thatŜ n for n ≥ 3 andÂ n for n ≥ 4 are simple polygroups. Also, we classify all subpolygroups ofD 2n and we show thatD 2n for even n, has exactly one normal subpolygroup and for odd n, is simple. This is a counter example that shows that the converse of our claim is not true in general. Also, we show that the polygroupsT 4n andÛ 6n are cyclic with finite period. Moreover, we prove that if G is a non-abelian simple group, thenĜ is a single power cyclic polygroup. Throughout this paper, χ 1 is the trivial character and for an irreducible character χ i , we denote χ i * χ i * ... * χ i t times by χ t i , where the hyperoperation * is as above.

PRELIMINARIES
In this section, we recall some definitions and facts about hypergroups and characters of finite groups, referring to [5,4] and [11].
Let Irr(G) = {χ 1 , χ 2 , ..., χ k }, where χ i for 1 ≤ i ≤ k are all complex irreducible characters of G. Then for any two characters χ i , χ j of G, (χ i , χ j ) denotes the usual inner product: (1) The row orthogonality relations: The column orthogonality relations:  .., χ a be the distinct irreducible characters of G and let ψ 1 , ..., ψ b be the distinct irreducible characters of H. Then G×H has precisely ab distinct irreducible characters, and these are and     Table 5. table of linear characters of T 4n for odd integer n  Irreducible characters of G containing K in their kernel are easily identified with the irreducible characters of G K . Thus we regard G K as a subset ofĜ and it is easily seen to be a subpolygroup.
K yields a one-one correspondence between the set of normal subgroups of G and the set of subpolygroups ofĜ. Now we summarize some basic facts a bout equivalence relations. Let H be a hypergroup and R ⊆ H × H be an equivalence relation on H. For non-empty subsets A and B of H, we define ARB ⇔ ∀a ∈ A, ∃b ∈ B such that aRb and ∀b ∈ B, ∃a ∈ A such that a Rb ; The relation R is called: (1) regular on the left (on the right) if xRy ⇒ a • xRa • y (x • aRy • a, respectively), for all x, y, a ∈ H.
(2) strongly regular on the left (on the right) if Moreover, R is called regular (strongly regular) if it is regular (strongly regular) on the right and on the left. (1) If R is regular, then H R is a hypergroup, with respect to the following operation:x ⊗ȳ = {z|z ∈ x • y}; (2) If the above operation is well defined on H R , then R is regular. (1) If R is strongly regular, then H R is a group, with respect to the following operation:x ⊗ȳ = {z|z ∈ x • y}; (2) If the above operation is well defined on H R , then R is strongly regular.
Let N be a normal subpolygroup of P . Then we define the relation x ≡ y(modN ) if and only if xy −1 ∩ N = φ. This relation is denoted by xN P y. Let N P (x) be the equivalence class of the element x ∈ P . Suppose that [P : N ] = {N P (x)|x ∈ P }. On [P : N ] we consider the hyperoperation defined as follows: For a subpolygroup K of P and x ∈ P , denote the right coset of K by Kx and let P K be the set of all right cosets of K in P .

SIMPLE CHARACTER POLYGROUPS
In this section, we classify all subpolygroups ofD 2n and we show that when n is an even integer,D 2n is not simple and for odd n it is simple. Also, we show that ifĜ is a single power cyclic polygroup, thenĜ is simple and in consequencê S n for n ≥ 3 andÂ n for n ≥ 4 are simple.
Theorem 3.1. The polygroupD 2n for even integer n = 2m has t+3 subpolygroups, where t is the number of divisors of n.
Proof. By Theorem 2.8, the set of subpolygroups ofD 2n is in one to one correspondence with the set of normal subgroups of D 2n . We know that the proper normal subgroups of D 2n are < a i >, < a 2 , b > and < a 2 , ab > where i | n. Now to obtain the subpolygroups N i related to < a i > consider two cases according to the parity of i and using Table 1: a) Suppose that i is an even integer. Then b) Suppose that i is an odd integer. Then The two remained subpolygroups ofD 2n are {χ 1 , χ 3 }, {χ 1 , χ 4 }. Therefore, all proper subpolygroups ofD 2n are obtained.
Theorem 3.2. The polygroupD 2n for an even integer n has exactly one normal subpolygroup.
Proof. We claim that N m is the only normal subpolygroup ofD 2n . Let N i = N m be a subpolygroup ofD 2n . By definition, N i is normal if for every ψ j , ψ j N i ψ j ⊆ N i . But the complex conjugate of ψ j is equal to ψ j and hence N i is normal if, ψ j 2 N i ⊆ N i . On the other hand, for each j according to Table 1 and using the orthogonality relations we have ψ j 2 = χ 1 + χ 2 + ψ k for some even integer k.
Also, ψ 1 2 = χ 1 + χ 2 + ψ 2 . Hence ψ 2 is a constituent of ψ 1 2 . But ψ 2 just belongs to subpolygroup N m , therefore N i is not normal. Now we show that N m is normal. By (1), ψ j 2 ⊆ N m and hence by Theorem 2.7, ψ j 2 N m ⊆ N m . Therefore, N m is normal. Proof. We know that the normal subgroups of D 2n are < a i > where i | n. Hence by Theorem 2.8 and Table 2, the subpolygroups ofD 2n are Definition 3.4. We say that the polygroup P is simple if it does not have any proper normal subpolygroups.
Theorem 3.5. The polygroupD 2n for odd integer n is simple.
Definition 3.6. For all n ≥ 1, we define the relation β n on a semihypergroup H, as follows: Suppose that β * is the transitive closure of β. Then β * is the smallest strongly regular relation on H [3]. In this case β * is called the fundamental equivalence relation on H. If H is a hypergroup, then β * = β.
Theorem 3.7. IfĜ is a single power cyclic polygroup, thenĜ is simple.
HenceĜ = (ψ) n−1 is a single power cyclic polygroup. Now using Theorem 3.7,Ĝ is a simple polygroup and the proof is complete.
where t ∈ N and n ≥ 4. Then the character polygroupĜ is a simple and single power cyclic polygroup.
Proof. Using Theorem 3.9, in the same manner as in Theorem 3.10, we can see thatĜ = (ψ ↓ G ) n−2 is a simple and single power cyclic polygroup.

CYCLIC CHARACTER POLYGROUPS
In this section, we prove that if G is a non-abelian simple group, thenĜ is a single power cyclic polygroup. Also, we show that the polygroupsT 4n andÛ 6n are cyclic with finite period.
Proposition 4.1. If G is a non-abelian simple group, thenĜ is a simple and single power cyclic polygroup.
Proof. By Theorem 2.8, we observe thatĜ is a simple polygroup. On the other hand, all irreducible characters of G are faithful and the order of G is an even integer. Thus G has a real class, and hence G has at least one real irreducible character χ. Then So we have χ 1 ∈ χ 2 ⊆ χ 4 ⊆ χ 6 ⊆ .... Since G is simple, χ 2 is faithful. Now suppose that χ 2 (g) takes on exactly m different values for g ∈ G. Then by Theorem 2.2, Therefore,Ĝ is single power cyclic.

Proof.
First, let n be an even integer. Using Tables 3 and ??, we prove that for 1 ≤ i ≤ 4, χ i is a constituent of ψ n 1 . Since n is an even integer, then ψ 1 n is a positive real character, and thus (ψ n 1 , χ j ) > 0 for j = 1, 2. Therefore, χ 1 , χ 2 ∈ ψ n 1 . Now for χ 3 we have: In the last equality we define numbers A k and B n−k for 0 ≤ k < n 2 as follow:
For all 0 ≤ k ≤ n 2 and k = n−j 2 we define numbers A k , A k , B n−k and B n−k as follow: Analysis similar to the case of even n, shows that for each k, And for k = n−j 2 , we have that each component of A k and B n−k is equal to one. Hence, For odd integer j, similarly, we can check that ψ j ∈ ψ 1 n−1 . Now let n be an odd integer. Using Tables 3 and 5, similar to above, we can show that χ 1 , χ 2 and ψ j for even integer j are constituents of ψ 1 n−1 and χ 3 , χ 4 and ψ j for odd integer j are constituents of ψ 1 n and the proof is complete.
Proof. We know that the character tables of S 3 and C n are as follow:   Table 9. character table of G = S 3 × C n
For other cases, the proof is the same as t = 1.
Corollary 4.4. Let n be an odd integer. Then U 6n ∼ = S 3 × C n , and thusÛ 6n is a cyclic polygroup with finite period.
Proof. It is clear that the mapping a → ((12), w) and b → ((123), 1) is an isomorphism between U 6n and S 3 × C n and the proof is complete.
When k is an odd integer, similar to above, we can prove that (ψ k , ψ 1 k ) = 2 k + 1 3 .
The proof is complete.

CONCLUSION
In this paper, we investigate some special polygroups in terms of simplicity and circularity and study the relation between single power cyclic polygroups and simple polygroups. These results raise the following problems: "Under what conditions a simple polygroup is going to be a single power cyclic polygroup? Is there a special class of finite groups for which the character polygroups are simple?"