ON CALCULATIONS OF MODULAR IRREDUCIBLE CHARACTERS WITH THE HELP OF COMPUTERS

KATSUSHI WAKI

Abstract. Let p be a prime. We denote p-modular system by (K, R, F) where F is an algebraically closed field of characteristic p. Let G be a finite group. It is not so easy to clculate of irreducible modular characters correspondent to simple F G-modules from ordinary characters correspondent to simple KG-modules. I will introduce a standard way written in [2] for the calculation of irreducible modular characters without a concrete construction of simple F G-modules. In the appendix, I also show a log of short demonstrations of the calculations of the decomposition matrix for the block of defect 2 in J_{4} by GAP[3].

1. Notation

For an element x ∈ G, o(x) denotes the order of x. Then we call

x : p-singular ↔ p|o(x)
x : p-regular ↔ p |̸o(x)

Let Irr(G) (IBr(G)) be a set of irreducible ordinary (Brauer) characters of G. Let C(G) be a set of all class functions of G. For simplification of description, for any φ ∈ IBr(G), φ(x) is defined 0 on p-singular elements x. So we can see that IBr(G) is a subset of C(G).

Let ℤ be a set of integers and ℕ be a set of natural numbers. Let X be a subset of C(G) then we define ⟨X⟩_{ℤ} be a set of all ℤ-linear combinations of X. We can also put ℕ instead of ℤ. For example, ⟨Irr(G)⟩_{ℕ} is a set of all ordinary characters of G and ⟨IBr(G)⟩_{ℕ} is a set of all Brauer characters of G.

It is well-known that Irr(G) (IBr(G)) is ℤ-basis of ⟨Irr(G)⟩_{ℤ} (⟨IBr(G)⟩_{ℤ})).

Moreover for an ordinary character χ, Let

χ̂(x) := \{ \begin{matrix}χ(x) & if x is p-regular \\ 0 & else \end{matrix}

Then χ̂ is in ⟨IBr(G)⟩_{ℕ}. In particular, we can find positive integers d_{χφ} for χ ∈ Irr(G) such that χ̂ = ∑_{φ∈IBr(G)} d_{χφ}·φ. We call the matrix \{d_{χφ}\}_{{χ∈Irr(G) \atop φ∈IBr(G)}} the decomposition matrix of

G. Thus if we get the decomposition matrix, we can get all irreducible Brauer characters. Let Φ_{φ} be the projective indecomposable character with respect to an irreducible Brauer character φ. Then IPr(G):={Φ_{φ}|φ ∈ IBr(G)} is a set of all projective indecomposable

characters of G. For λ, μ ∈ C(G), Scalar product ⟨λ, μ⟩ is \frac{1}{|G|} ∑_{x∈G} λ(x)\overline{μ(x)}. For any

φ ∈ IBr, Φ_{φ} = ∑_{χ∈Irr(G)} d_{χφ} · χ. Thus we can get the decomposition matrix if we get all

indecomposable projective characters in B.

2. Basic Set and System of Atom

In this section, we introduce two ℤ-basis of a set of ℤ-linear combinations of characters. We can get all irreducible Brauer characters from these ℤ-basis. We start to show the elementary theorem.

Theorem 2.1. For Φ_{φ} ∈ IPr(G) and ψ ∈ IBr(G),

⟨Φ_{φ}, ψ⟩ = δ_{φψ}

Next proposition is used to find characters which is NOT projective characters.

Proposition 2.2. For Φ ∈ ⟨Irr(G)⟩_{ℕ}, if Φ(x) = 0 (∀x : p-singuler ), and ⟨Φ, φ⟩ ≧ 0 (∀φ ∈ IBr(G) ) then Φ is a projective character.

Let B be a p-block of G. We denote Irr(B), IBr(B) and IPr(B) sets of characters of Irr(G), IBr(G) and Irr(G) in B, respectively. The matrix D_{B} = \{d_{χφ}\}_{{χ∈Irr(B) \atop φ∈IBr(B)}} is called

the decomposition matrix of B.

Let s be the number of irreducible ordinary characters in B. Let t be the number of irreducible Brauer characters in B.

Definition 2.3. Let BS and PS be ℤ-basis of ⟨IBr(B)⟩_{ℤ} and ⟨IPr(B)⟩_{ℤ}, respectively. We call BS is a basic set of IBr(B) if and only if BS ⊂ ⟨IBr(B)⟩_{ℕ} and PS is a basic set of IPr(B) if and only if PS ⊂ ⟨IPr(B)⟩_{ℕ}.

We denote M_{t}(ℤ) a set of t × t-matrix over ℤ.

Lemma 2.4. Let BS and PS be ℤ-basis of ⟨IBr(B)⟩_{ℤ} and ⟨IPr(B)⟩_{ℤ}. Let t × t-matrix U := ⟨BS, PS⟩ = {⟨φ, Φ⟩ | φ ∈ BS, Φ ∈ PS} Then both BS and PS are basic sets if and only if U^{−1} ∈ M_{t}(ℤ). In paticular, if U is the identity matrix, BS equals IBr(B) and PS equals IPr(B).

Definition 2.5. Let BA and PA be ℤ-basis of ⟨IBr(B)⟩_{ℤ} and ⟨IPr(B)⟩_{ℤ}, respectively. We call BA is a system of atom of IBr(B) if and only if ∀φ ∈ IBr(B); φ ∈ ⟨BA⟩_{ℕ} and PA is a system of atom of IPr(B) if and only if ∀Φ ∈ IPr(B); Φ ∈ ⟨PA⟩_{ℕ}.

Next lemma is used to check indecomposability of projective characters.

Lemma 2.6. For any Φ ∈ ⟨IPr(B)⟩_{ℕ}, if Φ ∈ PA then Φ ∈ IPr(B).

Definition 2.7. Let BS be a basic set of IBr(B). We call BS is special if any characters in IBr(B) are liftable s.t. ∀φ ∈ BS; ∃χ ∈ Irr(U) φ = χ on p-regular elements.

Last lemma help us to get outline of decomposition matrix.

Lemma 2.8. If B has a special basic set of IBr(B) then D_{B} can be like

( \begin{matrix}1 & & 0 \\ ∗ & 1 \\ ∗ & ∗ & ... \\ ∗ & ∗ & ∗ & 1 \\ ∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ & ∗ \end{matrix} ) .

3. Construction of BS, PS, BA and PA

3.1. How to make BS.

Lemma 3.1. Let S ⊂ ⟨IBr(B)⟩_{ℕ}. Then S is a basic set of IBr(B) if and only if ∀χ ∈ Irr(B), χ̂ ∈ ⟨S⟩_{ℤ} and S is linearly independent over ℤ.

The algorithm to get a basic set of IBr(B) is the following. Let Irr(B) = {χ_{1}, χ_{2}, …, χ_{s}}. Assume that we have a set of k Brauer characters B = {β_{1}, …, β_{k}} such that B is linearly independent over ℤ and there is a number r ≦ s such that {χ̂_{1}, …, χ̂_{r}} ⊂ ⟨B⟩_{ℤ}

Step 1 if r = s then B is a basic set and finish else goto Step 2. Step 2 if r < s and χ̂_{r+1} ∈ ⟨B⟩_{ℤ} then r := r + 1 and goto Step 1 else goto Step 3. Step 3 if χ̂_{r+1} ∉ ⟨B⟩_{ℚ} then add β_{k+1} := χ̂_{r+1} to B, r := r + 1, k := k + 1 and goto Step 1

else goto Step 4. Step 4 if χ̂_{r+1} ∈ ⟨B⟩_{ℚ} − ⟨B⟩_{ℤ} then

(i) Find the smallest integer m_{1} > 0 such that m_{1}χ̂_{r+1} = ∑_{i=1}^{k} z_{i}β_{i} ∈ ⟨B⟩_{ℤ}.

(ii) Let m be the minimum in {gcd(m_{1}, z_{j}) | 1 ≦ j ≦ k}.

(iii) Find a, b ∈ ℤ such that m = am_{1} + bz_{j} for a number j of m = gcd(m_{1}, z_{j}).

(iv) Let c_{i} be the minimum of {x ∈ ℕ | x + bz_{i}m_{1}^{−1} ≧ 0} for i ≠ j.

(v) Let β̃_{j} := bχ̂_{r+1} + aβ_{j} + ∑_{i≠j} c_{i}β_{i}

Then m_{1}β̃_{j} := mβ_{j} +m_{1} ∑_{i≠j}(c_{i}+bz_{i}m_{1}^{−1})β_{i} is in ⟨B⟩_{ℕ} and is a Brauer character. Thus

β̃_{j} is a Brauer character, too. So replace β̃_{j} as β_{j} in B then {χ̂_{1}, …, χ̂_{r+1}} ⊂ ⟨B⟩_{ℤ}. Let r := r + 1 and goto Step 1.

3.2. How to make PS.

Theorem 3.2. Let Mx be a set of all maximal subgroups of G and P_{Mx} := {ψ^{G} | ψ ∈ IPr(H), H ∈ Mx}. Then ⟨IPr(G)⟩_{ℤ} = ⟨P_{Mx}⟩_{ℤ}.

From this theorem, we can make PS from {Ψ_{B} | Ψ ∈ P_{Mx}} with the same way in 3.1. {χ̂ | χ ∈ Irr(B)}.

3.3. How to make BA and PA. Assume that we have succeed to make BS = {φ_{1}, … φ_{t}} and PS = {Φ_{1}, … Φ_{t}}. Find subset BA := {Φ_{1}^{∗}, …, Φ_{t}^{∗}} ⊂ ⟨IBr(B)⟩_{ℤ} such that ⟨Φ_{i}^{∗}, Φ_{j}⟩ = δ_{ij}. Find subset P A := {φ_{1}^{∗}, …, φ_{t}^{∗}} ⊂ ⟨IPr(B)⟩_{ℤ} such that ⟨φ_{i}^{∗}, φ_{j}⟩ = δ_{ij}. We can get the next lemma from Theorem 2.1.

Lemma 3.3. Above BA and PA are systems of atoms of IBr(B) and IPr(B).

Thus good BS and PS make better BA and PA. In the next section, we show that BA and PA make better BS ans PS, repectively. And the best BS and PS are IBr(B) and IPr(B) from lemma 2.4.

4. Indecmposablility check of projective characters

Definition 4.1. Let Φ be a projective character and Φ = ∑_{i=1}^{t} n_{i}φ_{i}^{∗} (n_{i} ≧ 0). We call Φ′

is a part of Φ if Φ′ = ∑_{i=1}^{t} n′_{i}φ_{i}^{∗} (0 ≦ n′_{i} ≦ n_{i}).

Theorem 4.2. ∀Φ′: a part of Φ; ∃φ : a Brauer character such that ⟨Φ′, φ⟩ < 0 or ⟨Φ − Φ′, φ⟩ < 0 then Φ is indecomposable.

Let Br (P s) be all Brauer (projective) characters which we got.

4.1. Subtract indecomposable projective characters from decomposable one. Let I be a suset of {1, …, t} such that ∀i ∈ I; Φ_{i}: indecomposable projective character.

Definition 4.3. For i ∈ I and j ∉ I, Let m_{ij} be the maximum of {n ∈ ℕ | ∀φ ∈ Br, ⟨φ, Φ_{j} − nΦ_{i}⟩ ≧ 0}. We call this m_{ij} a possible multiplicity of Φ_{i} in Φ_{j}.

Definition 4.4. Let i ∈ I and φ ∈ BS such that ⟨φ, Φ_{i}⟩ > 0 and φ = ∑_{i=1}^{t} n_{i}Φ_{i}^{∗} (n_{i} ≧ 0).

We call φ′ a bit of φ with respect to Φ_{i} if φ′ satifies the following conditions.

(a) φ′ = ∑_{i=1}^{t} n′_{i}Φ_{i}^{∗} (0 ≦ n′_{i} ≦ n_{i}).

(b) if j ∈ I then n′_{j} = δ_{ij} else n′_{j} ≦ m_{ij}.

(c) ∀Ψ ∈ P r, ⟨φ′, Ψ⟩ ≧ 0 and ⟨φ − φ′, Ψ⟩ ≧ 0.

Lemma 4.5. For i ∈ I, let β_{i} be the irreducible Brauer character corresponds to Φ_{i}. If φ ∈ BS and ⟨φ, Φ_{i}⟩ > 0 then β_{i} is a bit of φ with respect to Φ_{i}. In paticular, if φ has only one bit with respcet to Φ_{i} then this bit is the irreducible Brauer character corresponds to Φ_{i}.

For i ∈ I and j ∉ I,

m(i, j, φ) := min {⟨φ′, Φ_{j}⟩ | φ′ is a bit of φ with respect to Φ_{i}}

Theorem 4.6. For i ∈ I and j ∉ I, Φ_{j} − Max {m(i, j, φ) | φ ∈ BS, ⟨φ, Φ_{i}⟩ > 0} Φ_{i} is a projective character.

Example

( Calculation of the decomposotion matrix of the defect 2 block in J_{4} )

GAP4, Version: 4.3fix4 of December 20, 2002, i686-pc-linux-gnu-gcc
gap> Read("BrauerCT2.g");
gap> ct:=CharacterTable("J4");;
gap> sub_ct:=List(Maxes(ct){[1,2]},CharacterTable);;
gap> BB:=FindProjectiveCharacters(ct,sub_ct,3,20);;
This is 1 th - try.
This is 2 th - try.
gap> DisplayBlocksInfo(BB[1]);
There are 20 blocks.

+++ Block [J4]B1 of defect 3 has 14 irr. char. and 9 irr. Brauer char. +++
+++ Block [J4]B2 of defect 3 has 14 irr. char. and 9 irr. Brauer char. +++
+++ Block [J4]B3 of defect 2 has 9 irr. char. and 5 irr. Brauer char. +++
+++ Block [J4]B4 of defect 1 has 3 irr. char. and 2 irr. Brauer char. +++
+++ Block [J4]B5 of defect 1 has 3 irr. char. and 2 irr. Brauer char. +++
+++ Block [J4]B6 of defect 1 has 3 irr. char. and 2 irr. Brauer char. +++
+++ Block [J4]B7 of defect 1 has 3 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B8 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B9 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B10 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B11 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B12 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B13 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B14 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B15 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B16 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B17 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B18 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B19 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
+++ Block [J4]B20 of defect 0 has 1 irr. char. and 1 irr. Brauer char. +++
gap> B2A:=ShallowCopy(BB[1][3]);;
gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 16 proj. characters. +++
 Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
   1: [  1  .  .  .  1  1  .  .  1] :  (*)
   2: [  .  1  1  .  1  1  2  2  2] :
   3: [  .  .  1  3  3  3  4  4  7] :
   4: [  .  .  1  2  2  2  3  3  5] :
   5: [  .  .  .  1  .  1  .  1  1] :  (*)
   6: [  .  .  .  1  3  1  3  1  4] :
   7: [  .  .  1  2  3  2  4  3  6] :
   8: [  .  .  2  1  4  1  6  3  7] :
   9: [  .  .  3  2  3  2  6  5  8] :
  10: [  1  6  5  3  2 10  6 14 10] :
  11: [  .  2  5  6  5  8 10 13 16] :
  12: [  .  3  6  5  4  8 10 14 15] :
  13: [  1  3  4  5  5  9  8 12 14] :
  14: [  .  6  7  3  3  9 10 16 13] :
  15: [  .  2  8  5  4  7 12 15 17] :
  16: [  2  6  5  4  7 12 10 15 16] :
gap> B2A.rank;
5
gap> TryToSubstractIndecProjectiveCharacters(B2A,5,6,10);
1
gap> AddProjRec(B2A,LinearCombOfProj([1,-1],B2A.proj{[6,5]}),false);;

gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 16 proj. characters. +++
Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
1: [  1  .  .  .  1  1  .  .  1] :  (*)
2: [  .  1  1  .  1  1  2  2  2] :
3: [  .  .  1  2  3  2  4  3  6] :
4: [  .  .  1  2  2  2  3  3  5] :
5: [  .  .  .  1  .  1  .  1  1] :  (*)
6: [  .  .  .  .  3  .  3  .  3] :
7: [  .  .  .  1  3  1  3  1  4] :
8: [  .  .  2  1  4  1  6  3  7] :
9: [  .  .  3  2  3  2  6  5  8] :
10: [  1  6  5  3  2 10  6 14 10] :
11: [  .  2  5  6  5  8 10 13 16] :
12: [  .  3  6  5  4  8 10 14 15] :
13: [  1  3  4  5  5  9  8 12 14] :
14: [  .  6  7  3  3  9 10 16 13] :
15: [  .  2  8  5  4  7 12 15 17] :
16: [  2  6  5  4  7 12 10 15 16] :
gap> AddProjRec(B2A,LinearCombOfProj([1/3],B2A.proj{[6]}),false);;
gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 14 proj. characters. +++
Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
1: [  1  .  .  .  1  1  .  .  1] :  (*)
2: [  .  1  1  .  1  1  2  2  2] :
3: [  .  .  1  2  2  2  3  3  5] :
4: [  .  .  .  1  .  1  .  1  1] :  (*)
5: [  .  .  .  .  1  .  1  .  1] :  (*)
6: [  .  .  2  1  4  1  6  3  7] :
7: [  .  .  3  2  3  2  6  5  8] :
8: [  1  6  5  3  2 10  6 14 10] :
9: [  .  2  5  6  5  8 10 13 16] :
10: [  .  3  6  5  4  8 10 14 15] :
11: [  1  3  4  5  5  9  8 12 14] :
12: [  .  6  7  3  3  9 10 16 13] :
13: [  .  2  8  5  4  7 12 15 17] :
14: [  2  6  5  4  7 12 10 15 16] :
gap> TryToSubstractIndecProjectiveCharacters(B2A,5,3,10);
2
gap> AddProjRec(B2A,LinearCombOfProj([1,-2],B2A.proj{[3,5]}),false);;
gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
4 th proj. char. in PS is indec. because it is in PA.
5 th proj. char. in PS is indec. because it is in PA.

gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 12 proj. characters. +++
Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
1: [  1  .  .  .  1  1  .  .  1] :  (*)
2: [  .  1  1  .  1  1  2  2  2] :
3: [  .  .  1  2  .  2  1  3  3] :
4: [  .  .  .  1  .  1  .  1  1] :  (*)
5: [  .  .  .  .  1  .  1  .  1] :  (*)
6: [  .  .  2  1  4  1  6  3  7] :
7: [  .  .  3  2  3  2  6  5  8] :
8: [  1  6  5  3  2 10  6 14 10] :
9: [  .  3  6  5  4  8 10 14 15] :
10: [  .  6  7  3  3  9 10 16 13] :
11: [  .  2  8  5  4  7 12 15 17] :
12: [  2  6  5  4  7 12 10 15 16] :
gap> TryToSubstractIndecProjectiveCharacters(B2A,4,3,10);
2
gap> AddProjRec(B2A,LinearCombOfProj([1,-2],B2A.proj{[3,4]}),false);;
gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 8 proj. characters. +++
Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
1: [  1  .  .  .  1  1  .  .  1] :  (*)
2: [  .  1  1  .  1  1  2  2  2] :
3: [  .  .  1  .  .  .  1  1  1] :  (*)
4: [  .  .  .  1  .  1  .  1  1] :  (*)
5: [  .  .  .  .  1  .  1  .  1] :  (*)
6: [  1  6  5  3  2 10  6 14 10] :
7: [  .  6  7  3  3  9 10 16 13] :
8: [  2  6  5  4  7 12 10 15 16] :
gap> TryToSubstractIndecProjectiveCharacters(B2A,3,2,10);
1
gap> AddProjRec(B2A,LinearCombOfProj([1,-1],B2A.proj{[2,3]}),false);;
gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 8 proj. characters. +++
Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
1: [  1  .  .  .  1  1  .  .  1] :  (*)
2: [  .  1  .  .  1  1  1  1  1] :
3: [  .  .  1  .  .  .  1  1  1] :  (*)
4: [  .  .  .  1  .  1  .  1  1] :  (*)
5: [  .  .  .  .  1  .  1  .  1] :  (*)
6: [  1  6  5  3  2 10  6 14 10] :
7: [  .  6  7  3  3  9 10 16 13] :

   8: [  2  6  5  4  7 12 10 15 16] :
gap> TryToSubstractIndecProjectiveCharacters(B2A,5,2,10);
1
gap> AddProjRec(B2A,LinearCombOfProj([1,-1],B2A.proj{[2,5]}),false);;
gap> SetBS(B2A);; SetPS(B2A);; SetBA(B2A);; SetPA(B2A);; SetPIM(B2A,10);;
gap> DisplayProjectiveCharacters(B2A);
+++ Block [J4]B3 of defect 2 has 5 proj. characters. +++
 Irr:   14 21 25 27 28 30 31 35 41  : Indec. Flag
----+-----+--+--+--+--+--+--+--+--+-:-------------
   1: [  1  .  .  .  1  1  .  .  1] :  (*)
   2: [  .  1  .  .  .  1  .  1  .] :  (*)
   3: [  .  .  1  .  .  .  1  1  1] :  (*)
   4: [  .  .  .  1  .  1  .  1  1] :  (*)
   5: [  .  .  .  .  1  .  1  .  1] :  (*)

References

[1] Dirk Eddelbuettel, The Quantian Scientific Computing Environment,

     (http://dirk.eddelbuettel.com/quantian.html)

.

[2] G. Hiss, C. Jansen, K. Lux, R. Parker, Computational Modular Character Theory,

     (http://www.math.rwth-aachen.de/~MOC)

.

[3] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3 ; 2002,

   (http://www.gap-system.org).

Department of Mathematical System Science Faculty of Science and Technology Hirosaki University Aomori, Hirosaki 036-8561 JAPAN

  E-mail address: waki@cc.hirosaki-u.ac.jp