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On a Six Dimensional Projective Representation of the Hall-Janko Group

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On a Six Dimensional Projective Representation of the Hall-Janko Group Pacific Journal of Mathematics ON A SIX DIMENSIONAL PROJECTIVE REPRESENTATION OF THE HALL-JANKO GROUP JOHN HATHWAY LINDSEY, II Vol. 35, No. 1 September 1970 PACIFIC JOURNAL OF MATHEMATICS Vol. 35, No. 1, 1970 ON A SIX DIMENSIONAL PROJECTIVE REPRESENTATION OF THE HALL- JANKO GROUP J. H. LlNDSEY, II It is shown that there is a unique group G with property /: G is a central extension of Z2 by the Hall-Janko group of order 604,800 in which a 7-SyIow subgroup S7 is normalized by an element of order four. Also, G has a six-dimensional complex representation X. The proof is rather round-about. First, it is shown that there are at most two six-dimensional linear groups X{G) projectively representing the Hall-Janko group, and all such linear groups are algebraically conjugate. The character table and generators found by M, Hall for G/Z(G) are used. It is shown that a linear group L over GF(9) coming from the one candidate for a six dimensional group projectively representing the Hall-Janko group actually satisfies property /. This is done by showing that L has a permutation representation on 100 three-dimensional subspaces of (GF(9))6 and the image is permutation isomorphic to HalΓs permutation group. Hall later studied the geometry of these subspaces. In the course of constructing the character table of any group G with property I, G was found to have a six- dimensional representation. Once this representation is known to exist, it is possible to give two easier ways of construct- ing generators. The faithful characters on G are given in the appendix with only one representative of each pair of algebraically conjugate characters. This paper fills a gap in [11] concerning a six-dimensional re- presentation X with character χ of a central extension G by the Hall-Janko group. In § 3, G is the subgroup of <?L(6, 9) coming from X(G) in § 2 taken modulo 3. In § 4, G is any group with property I. In all cases, G turns out to be the same group. By [11], §9, Z = Z{G) has order 2 and Q(χ(G)) = Q(VT). We let χ' be the algebraic conjugate of χ and let Z = <—1>. The characters ψt and Pi are the characters from character tables for the Hall-Janko group and U3(3) respectively, in [9]. For p a prime, we let Sp be a p-Sylow subgroup of G and let S7 = <7Γ7>. By [11], § 9, (7- ΐ)/t7 = 6 = [N(S7): C(S7)] and C(S7) = S7Z. We let β, ε, and w be primitive seventh, fifth, and third roots of unity, respectively. Also, i is an element of order four in GF(9) and ά means a taken modulo a prime ideal dividing 3. Con way, [6, p. 86], independently discovered this projective re- presentation from the existence of the Hall-Janko group as a section 175 176 J. H. LINDSEY, II in Conway's group. 2* Construction of the matrix group χ(G). Using the notation of [9], χχ' = ti and χχ = fQ + tu + either Ψ16 or f17. Let α, 6, c, and £ correspond to those elements in [9]. We may let X{a) = /0 1 0\ /0 1 0\ diag (β, β\ β\ β~\ β-\ /5-4), X(e) =0010001, and X(t) = \1 0 0/ \1 0 0/ f r or Since c^ac^tf, this corresponds to multiplying the permutations in [9] from right to left. The permutation represen- tation of G on 100 letters corresponds to ^0 + ^1 + ΊAY The permu- tation representation of H on 36 letters corresponds to ft+ft+ft + ft, e f where H p* U3(S), [9]. We may write X(b) = Σ Ae.f,gX(t )X(c )X(a*) with Σ over 0^ -e^6, 0^ -/^2, -l^^O. We may also insist that Σo<^6 A_e,f>g = 0 for all /and g. For 0 ^ F^ 6, 0 ^ i^ 2, and O^G^l; trace X(aEcFtGb) = trace * ^— E,-F,~G ~ 2^0^e^G^-e,-F,-G By [11], §11, a central extension of Z2 by £73(3) is trivial and G contains a subgroup H^ Z73(3) in which we take α, 6, and c. If G = 0, then aEcFtGb e ίZ" and its cycle structure (using the permutation E F G representation giving p0 + pQ + ft + ft) will determine x(a c t b) — £J F G trace [X(α c ί δ)] = 7A^Et^Ft^G. If G = 1, then the cycle structure (using the permutation representation giving Ψ\/ > + ΨΊ + 'ΨV) determines E F G the class of a e t b within permutation of 7ΓX and π2, and within a sign. Now if "t" is the element of order 3 in H with | CH("ί") I = 108, then 27 | | CG("Γ) | and "ί" corresponds to Γ, so X(Γ) has eigen- 1 values w, w, w, w, w, w. As C(T)/T & AQ and T is conjugate to T" , X|C(Γ) has two 3-dimensional constituents representing A6 projec- tively. Allowing confusion of χ with χ', χ(π) = l + ε2 + ε~2 + ε2 + ε-2 + l or l + ε + ε~1 + l + ε2+ε-2. It must be the former since χ(τr)χ(τr)' = ^(π) - -4 where ε' = ε2. Therefore, χ(π) = 2Θ, and χ(πT) = -^ = ( + 1 ± l/T)/2. By [9], C{TύK- Γi> ^ A4. As X| ^TJ has some two dimensional constituent with kernel contained in ^T^, by [1], C(T^jζT^ is iso- morphic to SL(2,3). Therefore, Jx is conjugate to —Jx in C^) and TiJi is conjugate to — TJγ. Similarly, C(π)/<X> is the nontrivial central extension of Z% by Aδ and πJγ is conjugate to —πJλ. There- fore, all faithful, irreducible characters of G vanish on Jιy T^u πjlf and πVi As Jι with eigenvalues i,i,i, —i,—i,—i is represented faithfully in each of the 2-dimensional constituents of C(π) ?& <V> x SL(2,5), there is an element πδeC(π) of order 5 with χ(τr5) or χ(πby = —3^ or —2^ — θ2. Clearly, π5 is not conjugate to π, and 2 τr5 - τrx or πξ. Since 1 - ψfa) = χ(π^χ(πy9 χ(π^ Φ - SO, Now χfa J) = r 1 + ^14(^1^) + t16ori7^iJ ) = 1 + θi and χ(τrxj) = ±θ<. Letting ^ = ±1 ON A SIX DIMENSIONAL PROJECTIVE REPRESENTATION 177 and θi = #lor2, and using the permutations and character tables in 2 [9]: becl(K) (write δ ~ K), 7AQy0y0 = 0; ab~K, 7i4LliO,o = 0; a b~K, 3 4 δ ±2 7A_2,0,0 = 0; α δ~c, 7A_3,0,0 = 0; α δ ~ α, 7A_4,0,0 = -1; a b ~ tb > 6 7A_6fo,o = l; α δ - c, 7A_6,o,o = 0; cδ - JfΓ, 7A0>-1,0 = 0; αcδ ~ α, 7A_w,0= 2 ±2 3 4 — 1; a cb ~ tb , 7A-2,-i,o = 1; α cδ ~ w, 7A-s,_1,0 = 2; a cb~a, 7A^^u0 = δ 6 2 — 1; a cb~c, 7A_5,_1,0 = 0; α cδ~α, 7A_βf_i,0 = — 1; c b~K, 7A0,_2,0 = 2 4 0; ac b~tb , 7A_1,_2,_0 = 1; αVδ~α, 7A_2,_2,0 = -1; αVδ~u, 7A_8,_8,0 = 5 2 2; a*c*b~a, 7A_4,_2,0 = -1; a c b~K> 7A_β,-2,o = 0; αVδ~α, 7A_6,_2,0 = 2 -1; ί6 - ±τrΓ, ±τr T, 7Ao,o,-i = δ^fl^; ±αί6 - Γ^, 7A^1,0,_1 = 0; 2 a tb ~ ±TR, 7A_2>(W = δo; a*tb ~ ±π*Jx (here i = 1 or 2 will be 4 5 assumed), 7A_3,o,-i = 0; α tf> - ±τrjj, 7A_4,0,_1 = δ_^_2; α ίδ - ±2Έ, β 7A_5,o,_1 = ^ α ίδ ~ ±α, 7A_β>0,-i = ^ ^ίδ - ±ττje/, 7Λ,-i,-i = ^0; 2 actb ~ ±a, 7A_1,_1,_1 = δ4; a ctb — ±JBΓ, 7A_2,_1,_1 = 0; cΛtfδ ^ 7Γe/"ri 4 5 7A_8,-if-i = 0; α cίδ - ±α, 7A_4,_1,_1 = δδ; α cίδ - ±π*Γ, 7A_B,-i,-i = i 2 δ_3^_3; a?ctb~±a, 7A_6,^1,_1 = δ6; c?ti>~±π T, 7A),_2,_1 = §A; αc ί& - ±R, 7A_1,_2,_1 = 2δ_4; αVίδ - ±π'Γ, 7A»lf_I,.1 - δ96>4; αVίδ - TΓV,, 7A_3,_2>_1 = 0; αVίδ ~ π*Jlf 7A_4>_2,_1 = 0; αVίδ - ±7rfJ, 7A_5,_2,_1 = Me; αVίδ - ±τr;, 7A_6,_2,_1 = δ_5(l + θ6). Making an arbitrary choice of ε and possibly conjugating by — 73073 and replacing t by — ί, we may take 0_L = 0X and cLL = —l Now 0 = ΣeAe,0,_i is rational and —Θ1 + 8_φ_2 = 0 or — 1. As there are 3 other terms —Θ1 + δ_2θ_2 is odd, and cL2#_2 = — θ2. Similarly, δsθ0 + <5_36L3 is odd, and δ_dθ^ = δ3θ'o. In 0 = Σ.-Aβ,-2,-n 2δ_4 and δ_5 cannot have the same sign or the 4 other ^ terms could not cancel the ±3. Therefore, we may let 2<5_4= -2<58 and <?_5(l + 06) = δB(l + θ6). There exists a matrix P such that P~1X(G)P is unitary. For all A in X(N(a)): P~ιA'P = P-'A-ψ = (P-'AP)-1 = (P-ιAF)' = P'A'(P')"1 and PP'A'iPP')-1 = A!. As X\N(a) is irreducible PP' = 1 ι ι ι al6 for some scalar α. Then for BeX(G), B' = P(P~ BP)- P~ = -1 = PP'B'(P')~ιP-1 - aI,B'cr% = B'. Therefore, X(G) is forced to be unitary when X(N(a)) is taken in normal form.
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