Author(s): Behnam Razzaghmaneshi
In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ Sn is said to be non-synchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number. When trying to check if some group is synchronizing, typically, we have only partial information about the graph but enough to say that it has an automorphism of some type, and the goal would be to have in hand a classification of the primitive groups containing permutations of that type. As an illustration of this, the key ingredient in some of the results in [2] was the observation that the primitive graph understudy has a 2-cycle automorphism and hence the automorphism group of the graph is the symmetric group. For many more examples of the importance of knowing the groups that contain permutations of a given type. This type of investigation is certainly very natural since it appears on the eve of group theory, with Jordan, Burnside, Marggraff, but the difficulty of the problem is well illustrated by the very slow progress throughout the twentieth century. Given the new tools available (chiefly the classification of finite simple groups), the topic seems to have new momentum. Let Sn denote the symmetric group on n points; a permutation g∈Sn is said to be imprimitive if there exists an imprimitive group contain in gg. An imprimitive group is said to beminimally imprimitiveifit contains no transitive proper subgroup. An imprimitive group G≤Sn is said to be maximally imprimitive if for all g Sn\G, the group?g, G ? is primitive. The next result, whose proof isstraightforward, provides some alternative characterizations of imprimitive permutations. Now in this paper we discuse Presentation for imprimitive second Subgroup of general linear group in dimention 2 over the field of pk-elements .
1.1. Definition: The Burnside poset P, of a group G is a poset with the following properties.
(i) Each element of P represents a conjugacy class of subgroups of G and each conjugacy class is represented exactly once in P.
(ii) If C and D are elements of P, then we write C<D if and only if at least one group in the conjugacy class represented by C is a subgroup of at least one group in the conjugacy class represented by D.
1.2. Definition: We draw the Burnside inclusion diagram of the Burnside poset P of a group G according to the following rules . (i) We represent elements of P by small black discs. (ii) If an element of P represents a conjugacy class cantaining a single group (which is therefore normal in G) , we draw a circle around the disc representing that element. (iii) We lable each disc with either the isomorphism type or the order of the groups in the conjugacy class it represents. (iv) We represent the relation C<D by placing the disc representing C lower on the page then the disc reperesenting D and by drowing a line between those two discs. Howerer , we suppress inclusionthat are implied by the reflexiveness and transitivity of <.
1.3. Definition: Let m > 2 and n > 3 we will denote by I 2 2m (the I standing for‘ inversion’) any group isomorphic to the following soluble group of order
1.4. Definition: We will denote by SL(2,3) any group isomorphic to the following soluble group of order 24 :
1.5. Definition: We will denote by GL(2,3) any group isomorphic to the following soluble group of order 48 :
The Burnside inclusion diagram of GL(2,3) is pictured in Figure 1. This can be checked via CAYLEY, or in the tables of Neubuser (1967) where GL(2,3) has the number 48.49.
1.6. Definition: We will denote by BO (for binary octahedral) any group isomorphic to the following soluble group of order 48 :
The Burnside inclusion diagram of BO is pictured in Figure 2. This can be checked via CAYLEY, or in the tables of Neubuser(1967) where BO has the number 48.50.
1.7. Definition: We will denote by NS (for‘ninety-six’) any group isomorphic to the following soluble group of order 96 :
The Burnside inclusion diagram of NS is pictured in Figure 3.
In this section we discuse some properties of subgroups M3 and M 4 .
Recall that M4 is only defined when pk Ξ1(mod 4). We construct a generating set for M4 in this section by use of preceding methods. Let z be a generator for the scalar group, and define u and v by
For some scalars λ 1 , λ2 , μ1 and μ2 .
Let w be a primitve 4-th-root of unity in F, and let δ be an element of F such that
where σ is -1 or w, according as p k is congruent to 1 or 5 modulo 8 , respectively. Since b dosenot normalise < u,v > , it is more convernient to work < ωu, ωv > Setting x := ωμ y:= ωv then we have
It is easy to show that < a,b, x, y > is a minimal supplement to the scalar group. If pk 5(mod8), k p then it is the unique such supplement.
If pk Ξ1(mod8), k p then there is just one other minimal supplement to the scalar group; it is<ωa,b, x, y >, which is isomorphic to GL(2,3) .
Note that wilson (1972 , theorem 3.2 , p.36) also observed that M4 Splits over its Fitting subgroup when F has a square root of- 2 (that is, when p k Ξ 1 (mod8)), Also, we have that
Finally , we investigate the action of field automorphism on M 4
3.1. Theorem: Every automorphism of F , acting entry-wise on the elements of M4 , normalises M4 .
Proof: Let θ be an automorphism of F. By looking at the entries of the matrices in our generating set for M4 , it is clear that the effect of θ on M4 is determined by ωθ and δθ . It is easly to check that , in any case, xθ is ± x, yθ is ± y , aθ is ± ωθ and bθ is b or by .(and ofcourse zθ is some power of itself). Therefore θ normalises M4 .
