Abstract
The automorphism group of the unitary group has a maximal subgroup of the form of order 20643840. In this paper, Fischer-Clifford theory is applied to the split extension group to construct its character table. Also, class fusion from into the parent group is determined.
1. Introduction
The unitary simple group has outer automorphisms of orders 2, 3, and 6 and hence automorphism groups of the forms , , and exist for (see the ATLAS [1]). The reader is referred to [2] for more information about the construction of matrix representations for the covering and automorphism groups of . Recently in [3], a 3-local identification is given for the group and its automorphism groups , , and . Also, we found in the ATLAS that one of the 16 maximal subgroups of is a split extension group of the type of index 891 and has order 10321920. The group has automorphism groups of the form , , and which sit maximally inside the groups , , and , respectively.
The character table of is stored in the GAP Library [4], whereas the character table of and are not yet uploaded in GAP. In this paper, the unknown Fischer-Clifford matrices [5] of and its associated character table will be constructed. Readers are referred to [6] on a survey of Fischer-Clifford theory. It is interesting to note that our group is the stabilizer of a singular vector , where is the irreducible module of dimension 20 over the field for (see [3]). The method of coset analysis as discussed in [7–9] will be used in the computation of the conjugacy classes of elements of the group . The Fischer-Clifford matrices and character table of were determined in [10].
2. Group
In this section, using a suitable permutation representation of , we identify our group as a split extension by with the aid of GAP [4] and MAGMA [11]. Then with the help of MAGMA we represent as a matrix group of degree 9 over the Galois field . Since acts absolutely irreducibly on its natural module , a split extension of the form exists. Then we create as a subgroup of and show with the help of MAGMA that is indeed an isomorphic copy of .
We construct within GAP, using its smallest permutation representation of degree 672 found in the online ATLAS of Finite Group Representations [12]. Next, we use the GAP commands “MS≔ ConjugacyClassesMaximalSubgroups ()”, “A1≔MS”, and “Size(A1)” to represent as a permutation group on 672 points and then use this permutation representation to construct within MAGMA. Using the sequence of MAGMA commands “a,b≔ChiefSeries(A1)”, “N≔ ”, “NormalSubgroups(A1)”, “IsNormal(A1,a)”, “IsElementaryAbelian(N)”, “C≔ Complements(A1,N)”, “Order(C)”, “C meet N”, and “IsIsomorphic(C, )”, we verified that .
Having as a permutation group on 672 points, we use the MAGMA commands “M≔GModule(A1,N)” and "‘M:Maximal” to represent as a matrix group of degree 9 over the Galois field . Thus we obtain the matrix group having 14 conjugacy classes of elements, where and . The generators and of are as follows:
The MAGMA command “IsAbsolutelyIrreducible(G)” tells us that the action of the matrix group on its natural module is absolutely irreducible. Thus a split extension of the type does exist. Hence we can construct as a subgroup of such that and , where , , and . The generators of the matrix group of degree 10 over are as follows:
Since we can represent and as a matrix and permutation group, respectively, we use the MAGMA command “IsIsomorphic ()’’ to confirm that . Hence we can regard as the split extension .
3. The Conjugacy Classes of
In this section, the conjugacy classes of are computed using the technique of coset analysis and readers are encouraged to consult [8, 9] for a sound theoretical background on this technique.
Throughout the remainder of this chapter, let be a split extension of by , where is the vector space of dimension 9 over on which the linear group acts. Since is represented as a matrix group, we used the MAGMA commands “O≔ Orbits()”, “O”, “O”, “O”,O”, and “O” to compute the orbit lengths of the action of on . We obtain 4 orbits of lengths 1, 21, 210, and 280 and using the MAGMA commands “P1≔ Stabilizer(,O)”, “P2≔ Stabilizer(,O)”, “P3≔ Stabilizer(,O)”, and “P4≔ Stabilizer(,O)”, we are able to compute the corresponding point stabilizers , , which are subgroups of . With the aid of MAGMA and also checking the indices of the maximal subgroups of in the ATLAS, the structures of the stabilizers are identified as , , , and , where and are maximal subgroups of . We should note here that the group has two nonconjugate isomorphic maximal subgroups and , having the same structure . The stabilizer sits maximally in . Alternatively, we can use [13] to identify the structures of the groups . Since the action of on does not fix any nontrivial subspace of , we have that is an irreducible module for . We can readily verify this fact by using the MAGMA command “IsIrreducible()”.
Let be the permutation character of on the classes of . Then, from methods that were developed by Mpono [14], we obtain that = +++ = , where , , , and are the identity characters of the point stabilizers , , induced to . Note that the identity characters are identified with the permutation characters of acting on the classes of the point stabilizers . We found that = , = , = , and = . The permutation characters are written in terms of the ordinary irreducible characters of . Since we have the generators and for , we compute the character tables of and the ’s directly in MAGMA and use these tables together with the fusion maps of the stabilizers into , to compute and . The values of on the different classes of determine the number of fixed points of each in . The values of are listed in Table 1.
The values of enabled us to determine the number of orbits ’s, , which have fused together under the action of , for each class representative , to form one orbit . Mpono in [14] used the technique of coset analysis to develop Programmes A and B in CAYLEY [15] for the computation of the conjucacy classes of a split extension , where is an elementary abelian -group for a prime on which a linear group acts. Ali [16] adapted Programmes A and B to be used in MAGMA. Programme A computes the values of the , whereas Programme B determines the order of the elements for each conjugacy class in . We obtain that has exactly 49 conjugacy classes. The parameters and are defined in [14]. The centralizer order of each class of is computed using the formula . All the information involving the conjugacy classes of are listed in Table 2.
4. The Inertia Groups of
Since has four orbits on , then by Brauer’s Theorem [17] acts on with the same number of orbits. The lengths of the 4 orbits will be 1,, , and where + + = 511, with corresponding point stabilizers , , , and as subgroups of such that =1, =, = , and = . We generate as a permutation group on a set of cardinality 672 within MAGMA. Then the maximal and submaximal subgroups of are computed. Now, considering the indices of these subgroups in , the number of the classes of these subgroups, and also the fact that has 49 conjugacy classes, we deduce that the action of on has orbits of lengths 1, , , and with respective point stabilizers , , , and . Thus we obtain four inertia groups = , , for on Irr(). Alternatively, we can also determine the inertia factor groups if we let be the matrix group of dimension 9 over formed by the transpose of the generators of . Then the action of on the classes of is the equivalent of acting on . Then with the help of MAGMA or GAP, we can easily verify that the action of on has orbits of lengths 1, 21, 210, and 280 with corresponding point stabilizers , , , and . The structures of and have been identified by checking the indices of the maximal subgroups of in the ATLAS. The structure of was determined by direct computations in MAGMA. The groups , , and are constructed from elements within and the generators are as follows:(i) =, , and where(ii) =, , and where(iii) =, , adn where
For the purpose of constructing the character table of , we use the above generators of the ’s to compute their character tables.
5. The Fusion of , , and into
We obtain the fusions of the inertia factors , , and into by using direct matrix conjugation in and their permutation characters in of degrees 21, 210, and 280, respectively. MAGMA was used for the various computations. The fusion maps of , , and into are shown in Tables 3, 4, and 5.
6. The Fischer-Clifford Matrices of
Having obtained the fusions of the inertia factors into and the conjugacy classes of displayed in the format of Table 2, we can proceed to use the theory and properties discussed in [9] or [14] to help us in the construction of the Fischer-Clifford matrices of . Note that all the relations hold since is an elementary abelian group.
For example, consider the conjugacy class of . Then we obtain that has the following form with corresponding weights attached to the rows and columns:
We have , , , and by using Theorem 5.2.4 and property (e) of the Fischer-Clifford matrix (both found in [14]). Thus we obtain the following form:
By the orthogonality relations for columns and rows and remaining properties of the matrix found in Chapter 5 of [14], we obtain the desired Fischer-Clifford matrix of given below:
For each class representative , we construct a Fischer-Clifford matrix . These are listed in Table 6.
7. Character Table of
Having obtained the Fischer-Clifford matrices, the fusion maps of the ’s into , and the character tables of the inertia factors , we construct the character table of following the methodology discussed in Section 5.2 of [14]. For example, we calculate the partial character table of corresponding to the coset of . From the Fischer-Clifford matrix we obtain that
Let , , , and be the partial character tables of the inertia factors for the classes which fuse to . Then the partial character table of on the classes is given by
Similarly, the partial character table associated with each coset is computed. If necessary, we will restrict some characters of to , to ensure that each partial character table corresponding to a coset will give rise to the desired set .
The character table of will be partitioned row-wise into 4 blocks , , , and where each block corresponds to an inertia group = . Therefore , where , , , and . The character table of is shown in Table 7. The consistency and accuracy of the character table of have been tested by using the GAP code labelled as Programme E in [18].
The information about the conjugacy classes found in Table 2 can be used to compute the power maps for the elements of and then with the aid of Programme E in [18] we can verify that we obtained the unique -power maps listed in Table 8 for our Table 7.
8. The Fusion of into
Since is a maximal subgroup of of index 891, then the action of on the cosets of gives rise to a permutation character of degree 891. We deduce from the character table of found in GAP that , where , , , and are irreducible characters of of degrees 1, 22, 252, and 616, respectively.
We are able to obtain the partial fusion of into , using the information provided by the values of on the classes of and the power maps of and . Then, the technique of set intersections for characters (see [9, 14, 19]) is applied to restrict some ordinary irreducible characters of of small degrees to , to determine fully the fusion of the classes of into .
Let be the character afforded by the regular representation of . We obtain that = , where and . Then can be regarded as a character of which contains in its kernel such that If is a character of than we have that Here is the identity character of and is the restriction of to . We obtain that where and are the sums of the irreducible characters of which are in the same orbit under the action of on , for . Let , where . Then we obtain that
Hence and thereforewhere = .
We apply the above results to some of the irreducible characters of of small degrees, which in this case are = , = , = , = , = , and = . Their respective degrees are 22, 22, 231, 231, 440, and 440. For we calculate that Now + 21 + 210 + 280 = 22, since = 22. Since = 1, we must have that = 1 and = = 0. Note that does not have irreducible characters of degree 22. We obtain that = + if the partial fusion of into is taken into consideration. Similarly, for and we calculate that and Since the respective degrees of and are 231 and 440, we have to solve the equations (i) + 21 + 210 + 280 = 231 and (ii) + 21 + 210 + 280 = 440, separately. If we are taking into account the fact that the set (see Table 7) does not have any irreducible characters of degrees 231 and 440 and also that and , we deduce that the two sets of values and are the only possibilities that satisfy equation (i) and (ii), respectively, hence we obtained that = + and = + . Similar computations were carried out to restrict the characters , , and to and we found that = , = , and = + .
By making use of the values of , , , , , and on the classes of and the values of , , , , , and on the classes of together with the partial fusion, the complete fusion map of into is given in Table 9.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The second author acknowledges the financial support provided by the South African Department of Defense and the academic support from the 1st author towards the completion of his M.S. degree at the University of the Western Cape.