A NEW EXISTENCE PROOF OF JANKO’S SIMPLE GROUP J_{4}

G. D. COOPERMAN, W. LEMPKEN, G. O. MICHLER, AND M. WELLER

Introduction

Janko’s large simple sporadic group J_{4} was originally constructed by Benson, Conway, Norton, Parker and Thackray as a subgroup of the general linear group GL_{112}(2) of all invertible 112 × 112-matrices over the field GF (2) with 2 elements, see [1] and [13]. So far the construction of the 112-dimensional 2-modular irreducible representation of J_{4} is only described in Benson’s thesis [1] at Cambridge University. Furthermore, its proof is very involved.

In his paper [12] Lempken has constructed two matrices x, y ∈ GL_{1333}(11) of orders o(x) = 42, o(y) = 10, respectively, which describe a 1333-dimensional 11modular irreducible representation of J_{4}. These two matrices are the building blocks for the new existence proof for J_{4} given in this article.

In [17] the fourth author has used this linear representation of the finite group G = ⟨x, y⟩ to construct a permutation representation of G of degree 173 067 389 with stabilizer M = ⟨x^{3}, y, (x^{14})^{t}⟩, where t = (x^{14}y^{5})^{2}. His main result is described in section 2. It is based on a high performance computation on the supercomputers of the Theory Center of Cornell University and the University of Karlsruhe.

Using Weller’s permutation representation we show in Theorem 5.1 of this article that the group G = ⟨x, y⟩ is simple and has order

|G| = 2^{12} · 3^{3} · 5 · 7 · 11^{3} · 23 · 29 · 31 · 37 · 43.

Furthermore, we construct an involution u_{1} ≠ 1 of G and an element a_{1} of order 3 as words in x and y such that H = C_{G}(u_{1}) has the following properties:

(a) The subgroup Q = O_{2}(H) is an extra-special group of order |Q| = 2^{13}
such that C_{H}(Q) = ⟨u_{1}⟩.
(b) P = ⟨a_{1}⟩ is a Sylow 3-subgroup of O_{2,3}(H), and C_{Q}(P) = ⟨u_{1}⟩.
(c) H/O_{2,3}(H) ≅ Aut(M_{22}), the automorphism group of the Mathieu group
M_{22}, N_{H}(P) ≠ C_{H}(P) ≅ 6M_{22}, the sixfold cover of M_{22}.

Hence G ≅ J_{4} by Theorem A of Janko’s article [11].

In fact we give generators of these subgroups of H in terms of short words in x and y, see Theorem 5.1. Therefore all the assertions of this result can easily be checked by means of the computer algebra systems GAP or MAGMA without using the programs of [17].

In section 1 we determine the group structure of the subgroup M = ⟨x^{3}, y, (x^{14})^{t}⟩, where t = (x^{14}y^{5})^{2}. Proposition 1.3 asserts that M is the split extension of an elementary abelian group E of order 2^{11} by the simple Mathieu group M_{24}. By Proposition 1.4 the restriction of Lempken’s 1333-dimensional 11-modular representation V of G = ⟨x, y⟩ to M decomposes into two irreducible 11-modular representations W and S of dimensions dim_{F} W = 45 and dim_{F} S = 1288. From these data the fourth author has constructed the above mentioned permutation representation of G having degree 173067389 in [17].

Section 3 is devoted to determine the group structure of H = ⟨x^{7}, y^{5}, (x^{14})^{a}, (r_{1})^{b}⟩ of G, where a and b are suitably chosen elements of G described in Lemma 3.2. In Lemma 3.1 we construct an involution u_{1} ≠ 1 of G such that H ≤ C_{G}(u_{1}). In fact, we show in Proposition 3.3 that the group H has all the properties stated in assertions (a), (b) and (c) above.

In section 4 we study the fusion of the involutions of the stabilizer M in G. Proposition 4.1 asserts that G has 2 conjugacy classes of involutions (u_{1})^{G} and (w_{1})^{G}. Using this result and another high performance computation determining the number of fixed points of the involution u_{1} on the permutation module of degree 173 067 389 we prove in Proposition 4.2 that H = C_{G}(u_{1}). In the final section it is shown that G is a simple group. This is done in Theorem 5.1, which completes our existence proof of Janko’s simple group J_{4}.

Concerning our notation and terminology we refer to the Atlas [4] and the books by Butler [3], Gorenstein [7], Gorenstein, Lyons, Solomon [8] and Isaacs [10].

1. Lempken’s subgroup G = ⟨x, y⟩ of GL_{1333}(11)

Throughout this paper F denotes the prime field GF (11) of characteristic 11. Let V be the canonical 1333-dimensional vector space over F. In Theorem 3.16 and Remark 3.21 of [12] Lempken describes the construction of two 1333 × 1333matrices x, y ∈ GL_{1333}(11) of orders o(x) = 42 and o(y) = 10, which will become the starting data for the construction and new existence proof of Janko’s group J_{4}. Because of their size these matrices cannot be restated here, but they can be received by e-mail from eowmob@@exp-math.uni-essen.de.

Throughout this paper G = ⟨x, y⟩ is the subgroup of GL_{1333}(11)generated by the matrices x and y of orders 42 and 10, respectively. The following notations are taken from Lempken’s article [12]. There he considers the subgroup M = ⟨x^{3}, y, (x^{14})^{t}⟩ with t = (x^{14}y^{5})^{2} as well. However we cannot quote any result of [12] on the structure of the subgroup M, because Lempken assumes the existence of the simple Janko group J_{4}.

In this section we show that the subgroup M is a split extension of an elementary abelian normal subgroup E of order |E| = 2^{11} by the simple Mathieu group M_{24}. Furthermore, the module structure of the restriction of the 1333-dimensional representation of G to the subgroup M is determined.

The following notations are kept throughout the remainder of this article. Notation 1.1. In G = ⟨x, y⟩ ≤ GL_{1333}(11) define the following elements: r_{0} = yx^{21}y^{−1} r_{1} = x^{14}yx^{21}y^{−1}x^{−14} = (r_{0})^{x^{28}} r_{2} = y^{3}x^{21}y^{7} r_{3} = x^{14}y^{3}x^{21}y^{7}x^{−14} = (r_{2})^{x^{28}} v_{1} = y^{6}x^{21}y^{4} v_{2} = y^{8}x^{21}y^{2} v_{3} = y^{4}x^{21}y^{6} v_{4} = x^{21} w_{1} = [x^{6}, y^{5}] u_{1} = [x^{−6}w_{1}x^{6}, r_{1}] = (v_{1}r_{2})^{2} = [x^{−12}(y^{5}x^{6})^{2}(x^{21})^{y^{−1}x^{28}}]^{2} u_{2} = (v_{3}r_{0})^{2} u_{3} = u_{1}(v_{4}r_{0})^{2} u_{4} = (v_{3}r_{2})^{2} u_{5} = u_{4}(v_{4}r_{2})^{2} u_{6} = [x^{21}(x^{21})^{y}]^{2} s_{1} = y^{2}r_{1}y^{−2} = (r_{1})^{y^{8}} s_{2} = (x^{21})^{yx^{28}} = x^{14}y^{−1}x^{21}yx^{−14} d_{1} = [(r_{1})^{b}, s_{1}], where b = y^{−2}x^{−6} d_{2} = (x^{21})^{y} a_{1} = d_{1}x^{6}d_{1}x^{24}d_{1} t_{1} = s_{1}(r_{1})^{b}s_{1} t_{2} = (x^{21})^{y^{5}} q_{0} = r_{1}r_{3}d_{1}s_{1}(x^{6}y^{2})^{4} a_{3} = s_{1}(q_{0})^{3}s_{1}(q_{0})^{4}s_{1}s_{2} a_{6} = t_{1}(x^{14}y^{5})^{−2}x^{14}(x^{14}y^{5})^{2}t_{1} z = (x^{14})^{t}, where t = (x^{14}y^{5})^{2} Observe that the elements a_{1}, a_{3}, a_{6}, z ∈ G have order 3, and q_{0} ∈ G has order 7. All other elements are involutions of G. Lemma 1.2. Let L = ⟨x^{6}, y^{2}, z⟩ ≤ G = ⟨x, y⟩. Let T = ⟨x^{6}, y^{2}⟩, j = (x^{6}y^{2}x^{12})^{3}, and s = y^{6}(y^{2}x^{18})^{4}. Then the following assertions hold:

(a) T = ⟨x^{6}, y^{2}⟩ ≅ GL_{4}(2).
(b) r_{1} = (j)^{s} ∈ T.
(c) r_{0} = (r_{1})^{z^{2}} ∈ L.
(d) t_{2} = (r_{0})^{y^{6}}, r_{2} = (r_{0})^{y^{8}}, d_{2} = (r_{0})^{y^{2}} ∈ L.
(e) E_{2} = ⟨r_{0}, r_{2}, d_{2}, t_{2}⟩ is an elementary abelian normal subgroup of
N = ⟨T, E_{2}⟩ with order |E_{2}| = 2^{4}.
(f) N = E_{2}T, E_{2} ∩ T = 1, and N is perfect.
(g) |L : N| = 759, and |L| = 2^{10} · 3^{3} · 5 · 7 · 11 · 23.
(h) L ≅ M_{24}, the simple Mathieu group.

Proof. (a) By MAGMA the simple group GL_{4}(2) ≅ A_{8} has the following presentation with respect to the generators a and b of orders 5 and 7, respectively:

a^{5} = b^{7} = (ba^{3})^{4} = 1 ,
(b^{2}a)^{2} · b^{−1}ab^{−1}a^{3}b^{−1}a^{−1} = 1 ,
(∗) b^{3}aba^{3}b^{−3}ab^{−1}a^{2} = 1 ,
b^{3}a^{−1}b^{−1}a^{3}b^{−2}a^{−1}b^{−1}aba^{−1} = 1 ,
b^{2}aba^{−1}b^{−1}a^{3}b^{−1}ab^{−1}aba = 1 ,
(b^{2}ab^{−1}a^{−1}b^{−1}a^{−1})^{2} = 1.

Choosing a = y^{2} and b = x^{6} it follows by means of MAGMA that all the relations of (∗) are satisfied. Hence T = ⟨x^{6}, y^{2}⟩ ≅ GL_{4}(2).

(b) Certainly j = (x^{6}y^{2}x^{12})^{3}, s = y^{6}(y^{2}x^{18})^{4} ∈ T. Hence (j)^{s} ∈ T. Using MAGMA one checks that r_{1} = (j)^{s}.

(c) From r_{1} ∈ T ≤ L we obtain (r_{1})^{z^{2}} ∈ L. Using MAGMA again one gets that r_{0} = (r_{1})^{z^{2}}.

(d) As y^{2} has order 5 it is easily checked that conjugation by y^{2} yields the following orbit:

y^{2} : t_{2} → r_{2} → r_{0} → d_{2} → r_{0}r_{2}d_{2}t_{2} → t_{2}.

Hence all assertions of (d) hold.

(e) Similarily x^{6} has the following conjugation action:

x^{6} : t_{2} → t_{2}, and r_{0} → r_{2} → d_{2} → r_{0}d_{2}

Thus E_{2} = ⟨r_{0}, r_{2}, d_{2}, t_{2}⟩ is a normal elementary abelian subgroup of N = ⟨T, E_{2}⟩ of order |E_{2}| = 2^{4}.

(f) As T is a simple group by (a) we now get N = E_{2}T, and E_{2} ∩ T = 1. Since E_{2} is a simple 2-modular representation of T, it follows that N is perfect.

(g) Using the coset enumeration algorithm of MAGMA we see that |L : N| = 759. Hence |L| = 2^{10} · 3^{3} · 5 · 7 · 11 · 23.

(h) A matrix computation inside GL_{1333}(11) shows that [z, t_{2}] = z. Therefore L = ⟨N, z⟩ is perfect by (f). As |L : N| is odd, any Sylow 2-subgroup of L is isomorphic to a Sylow 2-subgroup of N, and therfore to those of L_{5}(2) or M_{24}. Applying now Theorem 1 of Schoenwaelder [16] we get L ≅ M_{24}.

□

Proposition 1.3. Let M = ⟨x^{3}, y, z⟩ ≤ G = ⟨x, y⟩. Then the following assertions hold:

(a) E = ⟨u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, v_{1}r_{0}, v_{2}r_{2}, v_{3}d_{2}, v_{4}t_{2}, y^{5}⟩ is an elementary abelian

normal subgroup of M with order |E| = 2^{11}.

(b) L = ⟨x^{6}, y^{2}, z⟩ is a subgroup of M such that M = EL, E ∩ L = 1,

and L is isomorphic to the simple Mathieu group M_{24} acting irreducibly on E.

(c) |M| = 2^{21} · 3^{3} · 5 · 7 · 11 · 23.
(d) M is perfect.
(e) M has six conjugacy classes of involutions with representatives:

u_{1}, r_{0}, r_{1}r_{2}, w_{1}, u_{4}r_{0} and u_{4}r_{1}r_{2}.

Proof. (a) By Lemma 1.2 (b) the elements t_{2}, d_{2}, r_{0}, r_{2} are contained in L = ⟨x^{6}, y^{2}, z⟩ ≤ M = ⟨x^{3}, y, z⟩. The following equations are verified by means of MAGMA:

[y^{5}, t_{2}] = v_{4}t_{2}, [y^{5}, d_{2}] = v_{3}d_{2}, [y^{5}, r_{2}] = v_{2}r_{2}, [y^{5}, r_{0}] = v_{1}r_{0}, [v_{2}r_{2}, r_{0}] = u_{1}, [v_{3}d_{2}, r_{0}] = u_{2}, [v_{4}t_{2}, r_{0}] = u_{1}u_{3}, [v_{2}d_{2}, r_{2}] = u_{4}, [v_{4}t_{2}, r_{2}] = u_{4}u_{5}, and [v_{4}t_{2}, d_{2}] = u_{6}. Hence E = ⟨u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, v_{1}r_{0}, v_{2}r_{2}, v_{3}d_{2}, v_{4}t_{2}, y^{5}⟩ ≤ M.

Using the computer and MAGMA it is checked that the 11 involutions u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, v_{1}r_{0}, v_{2}r_{2}, v_{3}d_{2}, v_{4}t_{2} and y^{5} commute pairwise, and that they generate an elementary abelian normal subgroup of M with order |E| = 2^{11}.

(b) L = ⟨x^{6}, y^{2}, z⟩ is a simple subgroup of M by Lemma 1.2. Thus E∩L = 1, and EL is a subgroup of M. We claim that M = EL. Certainly, y = y^{5} · (y^{2})^{3} ∈ EL. Lemma 1.2 (d) asserts that t_{2} ∈ L. Hence

x^{3} = (x^{6})^{4} · x^{21} = (x^{6})^{4} · v_{4} = (x^{6})^{4} · (v_{4}t_{2}) · t_{2} ∈ EL

Thus M = ⟨x^{3}, y, z⟩ = EL. It is well known that the smallest, non-trivial, irreducible 2-modular representation of L ≅ M_{24} is of degree 11. Hence L acts irreducibly on E, because 1 ≠ w_{1} = [x^{6}, y^{5}] ∈ [L, E].

(c) By (a), (b) and Lemma 1.2 (g) we have

|M| = |E · L| = 2^{11} · 2^{10} · 3^{3} · 5 · 7 · 11 · 23 = 2^{21} · 3^{3} · 5 · 7 · 11 · 23.

(d) As L is a simple group, L = L′ ≤ M′. Since L acts irreducibly on E by (b), we have E = [E, L] ≤ M′. Therefore M = EL = M′.

(e) It is well known that the simple Mathieu group M_{24} has 2 non-isomorphic simple 2-modular representations of degree 11. They are dual to each other. Hence there are 2 non-isomorphic split extensions 2^{11}M_{24}. By GAP [15] the character tables of these groups are both known. It follows that M has six conjugacy classes. Certainly, u_{1}^{M}, w_{1}^{M}, r_{0}^{M}, (u_{4}r_{0})^{M} and (r_{1}r_{2})^{M} are 5 different conjugacy classes of involutions of M, because their lengths |u_{1}^{M}| = 7 · 11 · 23, |w_{1}^{M}| = 2^{2} · 3 · 23, |r_{0}^{M}| = 2^{4}·3^{2}·5·11·23, |(u_{4}r_{0})^{M}| = 2^{4}·3^{2}·5·7·11·23, and |(r_{1}r_{2})^{M}| = 2^{6}·3^{2}·7·11·23 are all distinct. Furthermore, |(u_{3}r_{1}r_{2})^{M}| = 2^{6} · 3^{2} · 7 · 11 · 23, but the matrices r_{1}r_{2} and u_{3}r_{1}r_{2} are not conjugate in M, because they have different traces tr(r_{1}r_{2}) = 9 and tr(u_{3}r_{1}r_{2}) = 0 in GF (11) as is checked by means of MAGMA. □

The following result is due to Lempken [12]. Proposition 1.4. Let G = ⟨x, y⟩, and M = ⟨x^{3}, y, z⟩. Then the following assertions hold:

(a) V = F^{1333} is a simple F G-module.
(b) M is a subgroup of G such that the restriction V_{|M} = W ⨁ S,

where W and S are simple F M-modules with dimensions dim_{F} W = 45 and dim_{F} S = 1288.

Proof. Assertion (a) is a restatement of Theorem 3.20 of [12]. (b) is checked by means of Parker’s Meat-Axe algorithm contained in GAP, see [15].

□

2. Transformation of G into a permutation group

In [5] Cooperman, Finkelstein, York and Tselman have described a method for the transformation of a linear representation of a finite group κ : X → GL_{n}(K) over a finite field K into a permutation representation π : X → S_{m}, where m denotes the index of a given subgroup U of X.

This transformation is an important idea, because most of the efficient algorithms in computational group theory deal with permutation groups, see [3]. In particular, there is a membership test for a permutation σ ∈ S_{m} to belong to the subgroup π(X).

Using Algorithm 2.3.1 of [6] M. Weller [17] strengthened the results of Cooperman et al. [5] as follows:

Theorem 2.1. Let K be a finite field of characteristic p > 0. Let U be a subgroup of a finite group X, and let V be a simple KX- module such that its restriction V_{|U} contains a proper non-zero KU- submodule W. Then there is an algorithm to construct:

(a) The stabilizer Û = Stab_{G}(W) = {g ∈ G|W g = W ≤ V },
(b) a full set of double coset representations x_{i}, 1 ≤ i ≤ k, of Û in G,

i. e. G = ⋃_{i=1}^{k}Ûx_{i}Û,

(c) a base [β_{1}, β_{2}, ⋯, β_{j}] and strong generating set {g_{s}|1 ≤ s ≤ q} of G with

respect to the action of G on the cosets of Û, which coincides with the given operation of G on the F U-submodule W of V.

Using an efficient implementation of this algorithm on the supercomputers of the Theory Center at Cornell University and of the computer center of Karlsruhe University M. Weller [17] has obtained the following result.

Theorem 2.2. Let G = ⟨x, y⟩ ≤ GL_{1333}(11) and M = ⟨x^{3}, y, z⟩. Then the following assertions hold:

(a) |G : M| = 173067389
(b) |G| = 2^{21} · 3^{3} · 5 · 7 · 11^{3} · 23 · 29 · 31 · 37 · 43
(c) If Ω denotes the index set of the cosets Mg_{i} of M in G, then G induces on

Ω = {1, 2, ⋯, 173067389} a faithful permutation action with stabilizer stab_{G}(1) = M.

(d) G = ⋃_{i=1}^{7}Mx_{i}M, where the double coset representatives x_{i} of M are given

by the following words:

x_{1} = 1  (|Mx_{1}M| = 1)
x_{2} = x^{14}c^{12}  (|Mx_{2}M| = 15180)

x_{3} = x_{6}cyc^{20}yc^{16}yc^{19}yc^{17}yc^{17}yc^{19}yc^{8}xc^{22}

= x_{6}y^{2}c^{16}yc^{13}yc^{4}yc^{10}yc^{18}xc^{21}  (|Mx_{3}M| = 28336)
x_{4} = x_{3}c^{3}x^{−1}c^{11}  (|Mx_{4}M| = 3400320)
x_{5} = x_{6}xc  (|Mx_{5}M| = 54405120)
x_{6} = xc^{12}xc^{3}xc^{2}  (|Mx_{6}M| = 32643072)
x_{7} = x_{5}xc^{9}xc^{16}  (|Mx_{7}M| = 82575360)

where c := (x^{14})^{t}y^{4}(x^{14})^{t}y^{−1}(x^{14})^{t} has order 23, and t = (x^{14}y^{5})^{2}.

3. Group structure of the approximate centralizer H

Lempken [12] determines a suitable involution u_{1} ∈ G and an approximation H of the centralizer C_{G}(u_{1}). It is now defined by means of the notation 1.1. Lemma 3.1. In G = ⟨x, y⟩ let a = r_{1}y^{−4}x^{6}y^{4} and b = y^{−2}x^{−6}.

The subgroup H = ⟨x^{7}, y^{5}, (x^{14})^{a}, (r_{1})^{b}⟩ of G = ⟨x, y⟩ contains the involution u_{1} ≠ 1, and

H ≤ C_{G}(u_{1}).

Proof. The subgroup H of G is defined in Lemma 2.5 of [12]. Using GAP [15] it can easily be checked that u_{1}^{2} = 1, and that u_{1} commutes with the given generators of H. □

The remainder of this section is devoted to determine the group structure of H.

Lemma 3.2. Let H = ⟨x^{7}, y^{5}, (x^{14})^{a}, (r_{1})^{b}⟩, M = ⟨x^{3}, y, z⟩, where a = r_{1}(x^{6})^{y^{4}} and b = y^{−2}x^{−6}. Let W = H ∩ M. Then the following assertions hold:

(a) |H : W | = 77
(b) Q = ⟨u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, v_{1}, v_{2}, r_{0}, r_{1}, r_{2}, r_{3}, d_{1}, s_{1}⟩ is a normal extra-special
2-subgroup of H with |Q| = 2^{13}.
(c) K = ⟨d_{2}, s_{2}, t_{2}, a_{1}, a_{3}, a_{6}⟩ is a subgroup of W with center Z(K) = ⟨a_{1}⟩
such that K ≅ 3A_{6}.
(d) A = ⟨u_{1}, u_{6}, v_{3}d_{2}, v_{4}t_{2}, y^{5}⟩ is an elementary abelian subgroup of W with
order |A| = 2^{5} normalized by K⟨t_{1}⟩, and A ∩ K⟨t_{1}⟩ = 1.
(e) H ∩ M = QAK⟨t_{1}⟩, and K⟨t_{1}⟩ ≅ 3S_{6}.
(f) |H| = 2^{21} · 3^{3} · 5 · 7 · 11

Proof. (a) By Theorem 2.2 G = ⟨x, y⟩ has a faithful permutation action on the 173067389 cosets Mg of the subgroup M = ⟨x^{3}, y, z⟩. Using the computer we restrict this permutation representation to the subgroup H. It follows that

|H : H ∩ M| = 77.

(b) Let Q = ⟨u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, v_{1}, v_{2}, r_{0}, r_{1}, r_{2}, r_{3}, d_{1}, s_{1}⟩. Using again the computer and the permutation representation of G described in Theorem 2.2 it follows that Q ≤ H ∩ M. Furthermore, we get the following relations: (u_{2})^{s_{1}} = u_{1}u_{2}, (u_{3})^{r_{3}} = u_{1}u_{3}, (u_{4})^{d_{1}} = u_{1}u_{4} (u_{1}u_{4}u_{5})^{r_{1}} = u_{1}(u_{1}u_{4}u_{5}) = u_{4}u_{5}, (v_{1})^{r_{2}} = u_{1}v_{2}, (v_{2})^{r_{0}} = u_{1}r_{0}.

Since u_{1} commutes with all the generators it follows that ⟨s_{1}, u_{2}⟩, ⟨r_{3}, u_{3}⟩, ⟨d_{1}, u_{4}⟩, ⟨r_{1}, u_{1}u_{4}u_{5}⟩, ⟨r_{2}, v_{1}⟩ and ⟨r_{0}, v_{2}⟩ are six dihedral subgroups of order 8 with almagamated subgroup ⟨u_{1}⟩, which commute pairwise as subgroups. Hence Q is their central product. In particular, Q is an extra-special 2-group of order |Q| = 2^{13}. Another matrix computation shows that Q is invariant under conjugation by the 4 given generators of H. Thus Q is normal in H.

(c) Let K := ⟨a_{1}, d_{2}, s_{2}, t_{2}, a_{3}, a_{6}⟩. Using the computer again we see that K is a subgroup of W = H ∩ M. Define a := t_{2}a_{1}^{2}d_{2}a_{3}^{2}a_{6}a_{3}^{2}a_{6}^{2}s_{2}a_{1}s_{2}a_{3}, b := s_{2}t_{2}, o(a) = 2, o(b) = 4 and o(ab) = 15.

Let x_{1} := b^{2}ab^{2}abab^{2}abab, x_{2} := a, x_{3} := b^{3}abab^{3}ab^{2}abab^{2}ab^{2}ab, x_{4} := b^{3}ab^{3}ab^{3}ab^{2}ab^{2}abababa, and c := x_{3}^{2}. Then the following relations hold in B = ⟨x_{1}, x_{2}, x_{3}, x_{4}⟩ ≤ K :

c^{3} = 1, x_{1}^{3} = 1, x_{2}^{2} = 1, x_{3}^{2} = c, x_{4}^{2} = 1, (x_{1}x_{2})^{3} = 1, (x_{2}x_{3})^{3} = 1, (x_{3}x_{4})^{3} = 1, (x_{1}x_{3})^{2} = 1, (x_{1}x_{4})^{2} = 1, (x_{2}x_{4})^{2} = 1, c^{x_{1}} = c, c^{x_{2}} = c, c^{x_{3}} = c, c^{x_{4}} = c. By Huppert [9], p.138 B/⟨c⟩ is isomorphic to the alternating group A_{6}. Thus |B| = 3|A_{6}|.

Now | ⟨a, b⟩ | = 3|A_{6}| by MAGMA. Furthermore we have a = x_{2}, and

b = x_{1}^{2}x_{2}x_{1}^{2}x_{3}^{4}x_{1}^{2}x_{2}x_{1}^{2}x_{3}^{4}x_{2}x_{4}x_{1}x_{4}x_{3}^{5}x_{2}x_{4}x_{3}^{2}x_{1}x_{3}^{5}x_{2}x_{3}x_{4}x_{1}^{2}x_{4}x_{3}^{2}x_{1}x_{2}x_{1}x_{4}x_{3}^{2}x_{4}x_{1}.

Hence B = ⟨x_{1}, x_{2}, x_{3}, x_{4}⟩ = ⟨a, b⟩. We claim that K = ⟨a, b⟩. This follows immediately from the following equations:

a_{1} = (bab^{2}ab^{3}abab^{3}ab^{3})^{8}
d_{2} = b^{2}
s_{2} = (bab^{2}ab^{3}abab^{3}ab^{3})^{6}
t_{2} = bab^{2}ab^{3}(ababab^{3})^{5}abab^{3}a
a_{3} = bab^{2}ab^{3}(ababab^{3})^{2}abab^{3}ab^{3}ababab^{3}
a_{6} = bab^{2}ab^{3}(ababab^{3})^{2}abab^{3}ab^{2} .

Let p_{1} = b^{2}ab^{2}abab^{2}ab^{3}abab^{3}. Then p_{1} has order 3 and commutes with a_{1}. Furthermore, [p_{1}, a_{3}] = a_{1}. Hence D = ⟨p_{1}, a_{3}⟩ is a Sylow 3-subgroup of K. It is extra-special. Therefore its center Z(D) = ⟨a_{1}⟩ = ⟨c⟩ does not split off. Hence K = ⟨a, b⟩ ≅ 3A_{6}, the non-split 3-fold cover 3A_{6} of A_{6}.

(d) Another application of the permutation representation of G described in Theorem 2.2 on the computer shows that A = ⟨u_{1}, u_{6}, v_{3}d_{2}, v_{4}t_{2}, y^{5}⟩ ≤ W, and that a_{1}, a_{3}, a_{6} ∈ W. Then A is an elementary 2-subgroup of W of order |A| = 2^{5}. Another computation shows that A is normalized by K⟨t_{1}⟩, and A ∩ K⟨t_{1}⟩ = 1.

(e) and (f) As (a_{1})^{t_{1}} = (a_{1})^{2}, (AK)^{t_{1}} = AK and K⟨t_{1}⟩ ≅ 3S_{6} by (c).

Certainly H ∩ M ≤ C_{M}(u_{1}) by Lemma 3.1. Using MAGMA we see that |(u_{1})^{M}| = 1771 = 7 · 11 · 23. Therefore |C_{M}(u_{1})| = |M| : |(u_{1})^{M}| = 2^{21} · 3^{3} · 5 by Proposition 1.3.

Thus H ∩ M = C_{M}(u_{1}), because |H ∩ M| ≥ |QAK⟨t_{1}⟩| = 2^{13} · 2^{4} · 2^{3} · 3^{3} · 5 · 2 = 2^{21} · 3^{3} · 5. Hence |H| = 2^{21} · 3^{3} · 5 · 7 · 11 by (a), and W = QAK⟨t_{1}⟩.

□

Proposition 3.3. Let H = ⟨x^{7}, y^{5}, (x^{14})^{a}, (r_{1})^{b}⟩ , where a = r_{1}(x^{6})^{y^{4}}, b = y^{−2}x^{−6}. Then the following assertions hold:

(a) U_{0} = ⟨u_{6}, v_{3}, v_{4}, d_{2}, s_{2}, t_{2}, a_{1}, a_{3}, a_{6}, x^{14}, y^{5}⟩ is a subgroup of H with center Z(U_{0}) = ⟨u_{1}a_{1}⟩

(b) U_{0}/Z(U_{0}) ≅ M_{22}, and U_{0} ≅ 6M_{22}

(c) Q ∩ Z(U_{0}) = ⟨u_{1}⟩.

(d) The element a_{1} of order 3 generates a Sylow 3-subgroup of O_{2,3}(H), and C_{Q}(a_{1}) = Z(Q) = ⟨u_{1}⟩.

(e) U = U_{0} : ⟨t_{1}⟩ = N_{H}(⟨a_{1}⟩).

(f) H = QU, U ∩ Q = U_{0} ∩ Q = Z(U) = ⟨u_{1}⟩, and U_{0} = C_{H}(a_{1})

(g) U/Z(U_{0}) ≅ Aut(M_{22}), the automorphism group of the simple Mathieu group M_{22}.

Proof. By Lemma 3.2 we know that u_{6}, v_{3}, v_{4}, s_{2}, t_{2} and d_{2} = (s_{2}t_{2})^{2} belong to W = H ∩ M. Certainly x^{14} ∈ H.

By Lemma 3.2 (e) we have a_{1}, a_{3} ∈ W. ¿From [y^{5}, s_{2}] = (y^{5}s_{2})^{2} = u_{1}u_{6} ∈ U_{0} follows that u_{1}a_{1} ∈ U_{0} and has order 6. By Lemma 3.2

U_{0} = ⟨u_{6}, v_{3}, v_{4}, d_{2}, s_{2}, t_{2}, a_{1}, a_{3}, a_{6}, x^{14}, y^{5}⟩

is a subgroup of H. Using MAGMA it is checked that the matrix u_{1}a_{1} commutes with the 11 generators of U_{0}. Thus ⟨u_{1}a_{1}⟩ ≤ Z(U_{0}).

By Lemma 3.2 (d) A = ⟨u_{1}, u_{6}, v_{3}d_{2}, v_{4}t_{2}, y^{5}⟩ is an elementary abelian subgroup of U_{0} with |A| = 2^{5}. By Lemma 3.2 (c) and (d) it is normalized by the perfect subgroup K = ⟨d_{2}, s_{2}, t_{2}, a_{1}, a_{3}, a_{6}⟩ of U_{0}. Using MAGMA it can be checked that A = [A, K] is a uniserial GF (2)K-module. Hence

AK = AK′ ≤ (U_{0})′.

Lemma 3.2 (a), (e) and (f) assert that |U_{0} : AK| = 77. Since x^{14} has order 3, and U_{0} = ⟨AK, x^{14}⟩ we get x^{14} ∈ U′_{0}. Hence U_{0} is perfect.

Let Ū_{0} = U_{0}/⟨u_{1}a_{1}⟩. Then Ū_{0} is perfect. Let H̄ = H/Q⟨a_{1}⟩. From Lemma 3.2 we get |H̄ : Ū_{0}| = 2, and |Ū_{0}| = 2^{7} · 3^{2} · 5 · 7 · 11.

Now we claim that Ū_{0} is simple. Since x^{14} does not normalize A, it follows that ⟨u_{1}a_{1}⟩ is the largest normal subgroup of U_{0} contained in AK. Thus O_{2}(Ū_{0}) = O_{3}(Ū_{0}) = O_{5}(Ū_{0}) = 1. Suppose that Y is a minimal normal subgroup of Ū_{0}. If |Y | is odd, then |Y | ∈ {7, 11}, and Ū_{0} splits over Y. Furthermore, Aut(Y) is cyclic. As Ū_{0} is perfect we get Y ≤ Z(Ū_{0}). Hence Ū′_{0} is a proper subgroup of Ū_{0}, a contradiction.

Therefore |Y | is even, and Y ∩ (Ā : K̄) ≠ 1. As Ā is not normal in Ū_{0}, we get Ā : K̄ ≤ Y. Thus Y = Ū_{0}, because |Ū_{0} : Y | is odd and Ū_{0} is perfect. Hence Ū_{0} is a simple group of order |Ū_{0}| = 2^{7} · 3 · 5 · 7 · 11. Now Theorem A of Parrott [14] asserts that Ū_{0} ≅ M_{22}, the simple Mathieu group M_{22}. Therefore (b) holds. Assertions (a) and (c) are immediately clear.

(d) Using MAGMA it can be seen that (a_{1})^{t_{1}} = (a_{1})^{2}. By Lemma 3.2 and (b) we have H = QU_{0}⟨t_{1}⟩, |H : QU_{0}| = 2 and O_{2,3}(QU_{0}) = Q⟨a_{1}⟩. Hence O_{2,3}(H) = Q : ⟨a_{1}⟩. Another computation with MAGMA yields that C_{Q}(a_{1}) = Z(Q) = ⟨u_{1}⟩.

(e) Certainly U = U_{0} : ⟨t_{1}⟩ ≤ N_{H}(⟨a_{1}⟩). In fact, U = N_{H}(⟨a_{1}⟩) by (d) and the Frattini argument applied to the Sylow 3-subgroup ⟨a_{1}⟩ of O_{2,3}(H).

(f) is now obvious.

(g) By Lemma 3.2 and (f) we know that (H ∩ M)/Q ≅ 2^{4} : ˆ3S_{6}. Therefore (H ∩ M)/O_{2,3}(H) ≅ 2^{4} : S_{6}. Hence H/O_{2,3}(H) ≅ Aut(M_{22}).

□

4. The order of C_{G}(u_{1})

In this section the order of the centralizer C_{G}(u_{1}) of the involution u_{1} ∈ G = ⟨x, y⟩ is determined. From Proposition 3.3 we then get: H = C_{G}(u_{1}). Proposition 4.1. The group G = ⟨x, y⟩ has two conjugacy classes of involutions with representatives u_{1}, w_{1} ∈ GL_{1333}(11) having traces tr(u_{1}) = 9, tr(w_{1}) = 0 ∈ F.

Proof. Certainly the matrices u_{1} and w_{1} are not conjugate in G, because they have different traces tr(u_{1}) = 9, tr(w_{1}) = 0 in F = GF (11). By Proposition 3.1 the following elements u_{1}, w_{1}, r_{0}, u_{4}r_{0}, r_{1}r_{2}, and u_{3}r_{1}r_{2} of M yield a complete set of representatives of all six conjugacy classes of M.

Let q_{0} = r_{1}r_{3}d_{1}s_{1}(x^{6}y^{2})^{4}, and a_{6} = (x^{14})^{y^{5}x^{14}}. Then in G the following fusion takes place:

u_{1} ∼ r_{0} ∼ (r_{1}r_{2}) and w_{1} ∼ (u_{4}r_{0}) ∼ (u_{3}r_{1}r_{2}),

because r_{0}^{y^{5}x^{14}r_{3}(q_{0})^{6}} = u_{1} = (r_{1}r_{2})^{y^{5}x^{14}t_{2}r_{2}s_{2}(q_{0})^{6}(a_{6})^{2}(q_{0})^{6}}, (u_{4}r_{0})^{y^{5}x^{14}r_{3}} = w_{1} = (u_{3}r_{1}r_{2})^{y^{5}x^{14}s_{2}(q_{0})^{6}y^{5}x^{14}r_{3}}

By Theorem 2.2 (b) the index of M in G is odd. Therefore each involution i of G has a G-conjugate i^{g} ∈ M. Hence i^{g} is contained in one of the six conjugacy classes of M. Since they are G-fused to u_{1}^{G} or w_{1}^{G} it follows that either i ∈ u_{1}^{G} or i ∈ w_{1}^{G}.

□

Proposition 4.2. H = C_{G}(u_{1})

Proof. Let f be the number of fixed points of the permutation afforded by u_{1} on the 173067389 cosets of M in G. As |G : M| = 173067389 is odd, each involution i of u_{1}^{G} is contained in f > 0 different conjugates M^{g} of M for some g ∈ G = ⟨x, y⟩. By the proof of Proposition 4.1 the group G fuses the conjugacy classes u_{1}^{M}, (r_{0})^{M} and (r_{1}r_{2})^{M}. Furthermore, |u_{1}^{M}| = 7 · 11 · 23, |r_{0}^{M}| = 2^{4} · 3^{2} · 5 · 11 · 23, and |(r_{1}r_{2})^{M}| = 2^{6} · 3^{2} · 7 · 11 · 23. Hence

|u_{1}^{G}| = \frac{|G : M|(|u_{1}^{M}| + |r_{0}^{M}| + |(r_{1}r_{2})^{M}|)}{f}
= \frac{173067389 · 11 · 23(7 + 2^{4} · 3^{2} · 5 + 2^{6} · 3^{2} · 7)}{f}
= \frac{173067389 · 11 · 23 · 4759}{f}

Using now the computer again, we see that u_{1} has f = 52349 fixed points. Therefore

|u_{1}^{G}| = |G : C_{G}(u_{1})| = 11^{2} · 23 · 29 · 31 · 37 · 43,

and |C_{G}(u_{1})| = 2^{21} · 3^{2} · 5 · 7 · 11 by Theorem 2.2. Now Lemma 3.2 and Proposition 3.3 assert that H = C_{G}(u_{1}). □

5. The main result

In this section we show that G = ⟨x, y⟩ is a simple group. As C_{G}(u_{1}) satisfies the hypothesis of Janko’s theorem A [11] by Propositions 3.3 and 4.2 our existence proof for Janko’s simple group J_{4} then is complete. Theorem 5.1. Let G = ⟨x, y⟩ where x, y ∈ GL_{1333}(11) are matrices constructed in [12] of orders o(x) = 42 and o(y) = 10. Then G is a simple group of order

|G| = 2^{21} · 3^{3} · 5 · 7 · 11^{3} · 23 · 29 · 31 · 37 · 43

such that u_{1} = [x^{−12}(y^{5}x^{6})^{2}(x^{21})^{y^{−1}x^{28}}]^{2} ≠ 1 is an involution of G with centralizer H = C_{G}(u_{1}) = ⟨x^{7}, y^{5}, (x^{14})^{a}, (r_{1})^{b}⟩, where a = r_{1}(x^{6})^{y^{4}}, and b = y^{−2}x^{−6} . Furthermore, with the Notation 1.1 the following assertions hold:

(a) Q = O_{2}(H) = ⟨u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, v_{1}, v_{2}, r_{0}, r_{1}, r_{2}, r_{3}, d_{1}, s_{1}⟩ is an extra-special normal subgroup of H with |Q| = 2^{13}.

(b) The element a_{1} = d_{1}x^{6}d_{1}y^{24}d_{1} of order 3 generates a Sylow 3-subgroup of O_{2,3}(H), and C_{Q}(a_{1}) = Z(Q) = ⟨u_{1}⟩.

(c) U_{0} = C_{H}(a_{1}) = ⟨u_{6}, v_{3}, v_{4}, d_{2}, s_{2}, t_{2}, a_{1}, a_{3}, a_{6}, x^{14}, y^{5}⟩ ≅ 6M_{22}, the sixfold cover of the

Mathieu group M_{22} with center Z(U_{0}) = ⟨u_{1}a_{1}⟩ .

(d) U = N_{H}(a_{1}) = U_{0} : ⟨t_{1}⟩ is a subgroup of H with U/Z(U_{0}) ≅ Aut(M_{22}), and center Z(U) = ⟨u_{1}⟩.

(e) H = QU, and Q ∩ U = ⟨u_{1}⟩ = C_{Q}(a_{1}). In particular, G is isomorphic to Janko’s simple group J_{4}.

Proof. In view of Proposition 4.2, Theorem 2.2, Lemma 3.2 and Proposition 3.3 it remains to show that G is a simple.

Proposition 3.1 asserts that M = ⟨x^{3}, y, (x^{14})^{t}⟩ = M′,where t = (x^{14}y^{5})^{2}. Hence x^{3}, y ∈ M′ ≤ G′. Furthermore, (x^{14})^{t} ∈ M′ ≤ G′. As G′ is normal in G we see that x^{14} ∈ G′. But gcd(3, 14) = 1 and so ⟨x⟩ = ⟨x^{3}, x^{14}⟩. Therefore G = ⟨x, y⟩ = G′, and G is perfect.

Let N be any normal subgroup of G. If |N| is even, then there is an involution y ≠ 1 in N. By Proposition 4.1 it is either conjugate to u_{1} or to w_{1} in G. Using Proposition 1.3 (b) and the fusion of the conjugacy classes u_{1}^{M}, r_{0}^{M}, (r_{1}r_{2})^{M} of involutions of M in G it follows that

⟨y^{G} ∩ M⟩ = M ≤ N.

By Theorem 2.2 the index |G : M| is odd. Hence G/N is a solvable group by the Feit-Thompson theorem. As G is perfect, we get N = G.

Therefore we may assume that |N| is odd. As u_{1} and u_{2} are two commuting involutions W = ⟨u_{1}, u_{2}⟩ is a Klein four-group acting on the normal subgroup N. Using the computer it follows that the matrix u_{1}u_{2} ∈ GL_{1333}(11) has trace tr(u_{1}u_{2}) = 9. Since tr(u_{1}u_{2}) = tr(u_{1}) Proposition 4.1 implies that all three involutions of W belong to u_{1}^{G}. Now the Brauer-Wielandt formula of [8], p. 198 asserts that

|N||C_{N}(W)|^{2} = |C_{N}(u_{1})|^{3}.

By Lemma 3.4 O_{2′}(H) = 1, because H = C_{G}(u_{1}) by Proposition 4.2. Hence C_{N}(u_{1}) = 1. Thus N = 1. Therefore G is a simple group, and G ≅ J_{4} by Theorem A of Janko [11]. □

ACKNOWLEDGEMENTS

The authors of this paper have been supported by the DFG research project ”Algorithmic Number Theory and Algebra”.

A substantial part of the high performance computations proving Theorem 2.2 were conducted using the resources of the Cornell Theory Center, which receives major funding from the National Science Foundation and New York State with additional support from the Research Resources at the National Insitutes of Health, IBM Cooperation and members of the Corporate Research Institute. The total computing time on all the involved notes was 28137 CPU-h. We owe special thanks to Professor J. Guckenheimer and Dr. A. Hoisie for their support.

The authors also would like to thank the Computer Center of Karlsruhe University for providing 38785 CPU-h on their supercomputer IBM RS/6000 SP with 256 knots. This help was necessary to complete the above mentioned computations. We are very grateful to Professor W. Schönauer for his assistance.

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College of Computer Science, Northeastern University, 161 Cullinane Hall, Boston, MA 02115, USA

E-mail address: gene@@ccs.neu.edu

Institut für Experimentelle Mathematik, Universität Essen-GH, Ellernstraße 29, Essen, Germany

E-mail address: lempken@@exp-math.uni-essen.de
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