  
  [1X2 [33X[0;0YClassical Groups and Aschbacher's Theorem[133X[101X
  
  
  [1X2.1 [33X[0;0YClassical Forms[133X[101X
  
  [33X[0;0YLet [23XV[123X be an [23Xn[123X-dimensional vector space over a finite field [23XK=\mathbb{F}_q[123X. A
  classical  form  on [23XV[123X is either a [23X\sigma[123X-sesquilinear form [23X\beta: V \times V
  \to  K[123X  (for some field automorphism [23X\sigma \in {\rm Aut}(K)[123X) or a quadratic
  form [23XQ: V \to K[123X.[133X
  
  
  [1X2.1-1 [33X[0;0YSesquilinear forms[133X[101X
  
  [33X[0;0YA  map  [23X\beta:  V  \times  V  \to  K[123X  is a [23X\sigma[123X-sesquilinear form if it is
  additive in both arguments and satisfies[133X
  
  
  [24X[33X[0;6Y\beta(\lambda u, \mu v) = \lambda \mu^\sigma \beta(u, v)[133X
  
  [124X
  
  [33X[0;0Yfor  all  [23Xu,  v  \in V[123X and [23X\lambda, \mu \in K[123X. When [23X\sigma = 1[123X (the identity
  automorphism),  [23X\beta[123X  is  simply  called [13Xbilinear[113X. The form is [13Xsymmetric[113X if
  [23X\beta(u, v) = \beta(v, u)[123X for all [23Xu, v[123X.[133X
  
  
  [1X2.1-2 [33X[0;0YQuadratic forms[133X[101X
  
  [33X[0;0YA  map  [23XQ:  V \to K[123X is a [13Xquadratic form[113X if [23XQ(\lambda v) = \lambda^2 Q(v)[123X for
  all [23Xv \in V[123X and [23X\lambda \in K[123X, and the associated [13Xpolar form[113X[133X
  
  
  [24X[33X[0;6Y\beta(u, v) := Q(u + v) - Q(u) - Q(v)[133X
  
  [124X
  
  [33X[0;0Yis  a symmetric bilinear form. Note that [23XQ[123X and [23X\beta[123X uniquely determine each
  other if [23X{\rm char}(K) \neq 2[123X.[133X
  
  
  [1X2.1-3 [33X[0;0YMatrix realizations of classical forms[133X[101X
  
  [33X[0;0YIn this package, classical forms on [23XV[123X are realized as (Gram) matrices.[133X
  
  [33X[0;0YFor a [23X\sigma[123X-sesquilinear form [23X\beta[123X, the Gram matrix [23XB[123X satisfies[133X
  
  
  [24X[33X[0;6Y\beta(u, v) = u B v^{\sigma {\rm T}}[133X
  
  [124X
  
  [33X[0;0Yfor all column vectors [23Xu, v \in V[123X. For a quadratic form [23XQ[123X, the Gram matrix [23XA[123X
  satisfies[133X
  
  
  [24X[33X[0;6YQ(v) = v A v^{\rm T}[133X
  
  [124X
  
  [33X[0;0Yfor all column vectors [23Xv \in V[123X.[133X
  
  [33X[0;0YFor  relevant  classical  forms, the package fixes a standard choice of Gram
  matrices in Subsection [14X2.2-6[114X.[133X
  
  
  [1X2.1-4 [33X[0;0YNon-degeneracy[133X[101X
  
  [33X[0;0YA sesquilinear form [23X\beta[123X is [13Xnon-degenerate[113X if its radical[133X
  
  
  [24X[33X[0;6Y{\rm Rad}(\beta) = \{v \in V \mid \forall u \in V: \beta(u, v) = 0 \}[133X
  
  [124X
  
  [33X[0;0Yis  trivial.  A  quadratic  form  is  non-degenerate  if  its  polar form is
  non-degenerate.[133X
  
  
  [1X2.1-5 [33X[0;0YIsometries and similarities[133X[101X
  
  [33X[0;0YLet [23Xg \in {\rm GL}(V)[123X. Then [23Xg[123X is an [13Xisometry of [23X\beta[123X[113X if[133X
  
  
  [24X[33X[0;6Y\forall u, v \in V: \beta(ug, vg) = \beta(u, v)[133X
  
  [124X
  
  [33X[0;0Yand an isometry of [23XQ[123X if [23XQ(vg) = Q(v)[123X for all [23Xv[123X. If instead equality holds up
  to a non-zero scalar [23X\lambda \in K^\times[123X, [23Xg[123X is a [13Xsimilarity[113X.[133X
  
  
  [1X2.1-6 [33X[0;0YClassification of sesquilinear forms[133X[101X
  
  [33X[0;0YLet [23X\beta[123X be a [23X\sigma[123X-sesquilinear form on [23XV[123X. Then following [BHR13, Theorem
  1.5.13],  we  focus  on  the  following  four classes of such forms, and the
  corresponding isometry groups:[133X
  
  [31X1[131X   [33X[0;6Y[23X\beta = 0[123X.[133X
  
  [31X2[131X   [33X[0;6Y[23X\sigma = 1[123X, [23X\beta(u, v) = -\beta(v, u)[123X for all [23Xu, v[123X, and [23X\beta(v, v) =
        0[123X for all [23Xv[123X. Then [23X\beta[123X is called [13Xalternating[113X or [13Xsymplectic[113X.[133X
  
  [31X3[131X   [33X[0;6Y[23X\sigma^2  = 1 \neq \sigma[123X and [23X\beta(v, u) = \beta(u, v)^\sigma[123X for all
        [23Xu, v[123X. Then [23X\beta[123X is called [13X[23X\sigma[123X-Hermitian[113X or [13Xunitary[113X.[133X
  
  [31X4[131X   [33X[0;6Y[23X\sigma  =  1[123X and [23X\beta(v, u) = \beta(u, v)[123X for all [23Xu, v[123X. Then [23X\beta[123X is
        [13Xsymmetric bilinear[113X.[133X
  
  [33X[0;0YIn  characteristic  2  the symplectic and symmetric cases overlap. Otherwise
  all cases are mutually exclusive.[133X
  
  
  [1X2.2 [33X[0;0YClassical Groups[133X[101X
  
  [33X[0;0YClassical  groups  in this package are realized as matrix groups over finite
  fields  that  preserve  a  specific  form on a vector space. These forms are
  represented  explicitly  by  their Gram matrices (with respect to a standard
  basis),  and  all  constructions  in the package are carried out relative to
  fixed choices of such matrices.[133X
  
  
  [1X2.2-1 [33X[0;0YLinear Groups (Case [22XL[122X[101X[1X)[133X[101X
  
  [33X[0;0YLinear  groups  preserve the zero form. We will denote the isometry group by
  [23X{\rm GL}_n(q)[123X.[133X
  
  
  [1X2.2-2 [33X[0;0YSymplectic Groups (Case [22XS[122X[101X[1X)[133X[101X
  
  [33X[0;0YSymplectic groups preserve non-degenerate symplectic forms. We will use [23X{\rm
  antidiag}(1,\dots,1,-1,\dots,-1)[123X  as our standard symplectic form matrix and
  denote the isometry group by [23X{\rm Sp}_n(q)[123X. Note that [23Xn[123X must be even.[133X
  
  
  [1X2.2-3 [33X[0;0YUnitary Groups (Case [22XU[122X[101X[1X)[133X[101X
  
  [33X[0;0YUnitary  groups  preserve  non-degenerate  unitary  forms, where [23X\sigma[123X is a
  field   automorphism   of   order   2.   So   they  are  only  defined  over
  [23X\mathbb{F}_{q^2}[123X   with   [23X\sigma:   x\mapsto   x^q[123X.   We   will   use   [23X{\rm
  antidiag}(1,\dots,1)[123X  as  our  standard  unitary  form matrix and denote the
  isometry group by [23X{\rm GU}_n(q)[123X.[133X
  
  
  [1X2.2-4 [33X[0;0YOrthogonal groups in odd dimension (Case [22XO[122X[101X[1X)[133X[101X
  
  [33X[0;0YIf  [23XQ[123X  is a quadratic form on [23X\mathbb{F}_q^n[123X with [23Xq[123X even and [23Xn[123X odd, then the
  isometry group of [23XQ[123X is isomorphic to a symplectic group of dimension [23Xn-1[123X.[133X
  
  [33X[0;0YFor this reason, we restrict attention to orthogonal groups of odd dimension
  over fields of odd characteristic. It is sufficient to define the polar form
  of  [23XQ[123X.  We  will  use [23X{\rm antidiag}(1,\dots,1,\frac{1}{2},1,\dots,1)[123X as our
  standard  non-degenerate  symmetric  bilinear  form  matrix  and  denote the
  isometry  group  by  [23X{\rm GO}_n(q)[123X. In odd dimension, there are two isometry
  classes of non-degenerate symmetric bilinear forms, distinguished by whether
  the  determinant  of  the  form  matrix  is  a  square  or  a  non-square in
  [23X\mathbb{F}_q^\times[123X. These two isometry classes, however, belong to the same
  similarity class.[133X
  
  
  [1X2.2-5 [33X[0;0YOrthogonal groups in even dimension (Cases [22XO^+[122X[101X[1X and [22XO^-[122X[101X[1X)[133X[101X
  
  [33X[0;0YIn  even  dimension,  non-degenerate  quadratic forms fall into two distinct
  isometry classes, which also correspond to two different similarity classes.
  Let  [23X\beta[123X  denote the polar form associated with a quadratic form [23XQ[123X. Over a
  field of odd characteristic, [23X\beta[123X and [23XQ[123X uniquely determine each other.[133X
  
  [33X[0;0YAssume  that [23Xq[123X is odd, let [23X\beta[123X be a non-degenerate symmetric bilinear form
  on [23X\mathbb{F}_q^n[123X with [23Xn[123X even. Then [23X\beta[123X is of [13Xplus-type[113X if it is isometric
  to  our  standard  form  matrix  [23X{\rm antidiag}(1,\dots,1)[123X, and otherwise of
  [13Xminus-type[113X.[133X
  
  [33X[0;0YIn  even characteristic, a non-degenerate quadratic form [23XQ[123X in even dimension
  is  of  plus-type  if  it  is  isometric  to  our  standard form matrix [23X{\rm
  antidiag}(1,\dots,1,0,\dots,0)[123X  and  of  minus-type otherwise. We denote the
  isometry group of a standard quadratic or bilinear form of plus-type by [23X{\rm
  GO}_n^+(q)[123X  and  the  isometry  group  of a standard form of minus-type (see
  [14X2.2-6[114X) by [23X{\rm GO}_n^-(q)[123X.[133X
  
  
  [1X2.2-6 [33X[0;0YStandard forms in [5XClassicalMaximals[105X[101X[1X[133X[101X
  
  [33X[0;0YLet [23XG[123X be the isometry group of one of the following forms on [23X\mathbb{F}_q^n[123X:
  the zero form, a unitary form, a symplectic form, a symmetric bilinear form,
  or a quadratic form, as described above.[133X
  
  [33X[0;0YThe  following  table  lists the corresponding form matrices preserved by [23XG[123X,
  which we adopt as our [13Xstandard form matrices[113X.[133X
  
  [33X[0;0YIn  odd  characteristic,  we specify quadratic forms via the matrix of their
  associated  polar  form  rather  than  the quadratic form itself. We use the
  notation [23X\nu=Z(q^2)[123X and [23X\xi=Z(q)[123X.[133X
  
        ────────────┬────────────────────┬─────────────┬───────────────────┬────────────────────────────────────────────────────────────  
           Case     │     conditions     │  form type  │    isom. grp.     │                            form                               
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
         [23X{\bf L}[123X  │         —          │    zero     │  [23X{\rm GL}_n(q)[123X  │                      [23X0_{n \times n}[123X                         
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
         [23X{\bf U}[123X  │         —          │  hermitian  │  [23X{\rm GU}_n(q)[123X  │                 [23X{\rm antidiag}(1,\dots,1)[123X                   
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
         [23X{\bf S}[123X  │      [23Xn[123X even      │ alternating │  [23X{\rm Sp}_n(q)[123X  │           [23X{\rm antidiag}(1,\dots,1,-1,\dots,-1)[123X             
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
         [23X{\bf O}[123X  │  [23Xq[123X odd, [23Xn[123X odd  │  symmetric  │  [23X{\rm GO}_n(q)[123X  │      [23X{\rm antidiag}(1,\dots,1,\frac{1}{2},1,\dots,1)[123X        
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
        [23X{\bf O}^+[123X │ [23Xq[123X odd, [23Xn[123X even  │  symmetric  │ [23X{\rm GO}^+_n(q)[123X │                 [23X{\rm antidiag}(1,\dots,1)[123X                   
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
        [23X{\bf O}^-[123X │ [23Xq[123X odd, [23Xn[123X even  │  symmetric  │ [23X{\rm GO}^-_n(q)[123X │ [23X{\rm antidiag}(1,\dots,1,-\nu-\nu^q,-\nu-\nu^q,1,\dots,1)[123X   
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
        [23X{\bf O}^+[123X │ [23Xq[123X even, [23Xn[123X even │  quadratic  │ [23X{\rm GO}^+_n(q)[123X │            [23X{\rm antidiag}(1,\dots,1,0,\dots,0)[123X              
        ────────────┼────────────────────┼─────────────┼───────────────────┼────────────────────────────────────────────────────────────  
        [23X{\bf O}^-[123X │ [23Xq[123X even, [23Xn[123X even │  quadratic  │ [23X{\rm GO}^-_n(q)[123X │     [23X{\rm antidiag}(1,\dots,1,-\nu-\nu^q,0,0,\dots,0)[123X        
        ────────────┴────────────────────┴─────────────┴───────────────────┴────────────────────────────────────────────────────────────  
  
       [1XTable:[101X Standard classical forms
  
  
  [33X[0;0YThese  are  exactly  the  forms  preserved  by  the groups returned by [5XGAP[105X's
  constructors [10XSL[110X, [10XSU[110X, [10XSp[110X, and [10XOmega[110X.[133X
  
  
  [1X2.3 [33X[0;0YAschbacher's Theorem[133X[101X
  
  [33X[0;0YAschbacher's  theorem is the organising principle behind the entire package.
  It  says  that  maximal  subgroups of finite classical groups fall into nine
  broad families:[133X
  
  [33X[0;0YLet  [23XG[123X be a quasisimple classical group acting naturally on a vector space [23XV[123X
  of  dimension  [23Xn  \geq  2[123X  over  [23X\mathbb{F}_q[123X and one of [23X{\rm SL}_n(q)[123X, [23X{\rm
  SU}_n(q)[123X, [23X{\rm Sp}_n(q)[123X or [23X\Omega^\varepsilon_n(q)[123X.[133X
  
  [33X[0;0YLet [23XH[123X be a maximal subgroup of [23XG[123X. Then one of the following holds:[133X
  
  [31X1[131X   [33X[0;6Y[23XH[123X  is  a  [13Xgeometric  subgroup[113X  and  belongs  to  one  of the [13Xgeometric[113X
        Aschbacher  classes [23X{\cal C}_1,\dots,{\cal C}_8[123X. These classes consist
        roughly of groups that preserve some kind of geometric structure on [23XV[123X.[133X
  
  [31X2[131X   [33X[0;6Y[23XH[123X  belongs to the [13Xnon-geometric[113X class [23X{\cal S}[123X (sometimes called [23X{\cal
        C}_9[123X). In this case,[133X
  
        [30X    [33X[0;12Y[23XH[123X is almost simple modulo scalars,[133X
  
        [30X    [33X[0;12Y[23XH^\infty[123X acts absolutely irreducibly,[133X
  
        [30X    [33X[0;12Ythere   does   not  exist  a  [23Xg  \in  {\rm  GL}_n(q)[123X  such  that
              [23X(H^\infty)^g[123X is defined over a proper subfield of [23X\mathbb{F}_q[123X,[133X
  
        [30X    [33X[0;12Y[23XH^\infty[123X  does not preserve a non-zero classical form on [23XV[123X other
              than those already preserved by [23XG[123X.[133X
  
  [33X[0;0YThe  following  table  gives  a  rough  description  of  the eight geometric
  Aschbacher classes, based on [KL90, Table 1.2.A].[133X
  
        ─────────────┬─────────────────────────────┬──────────────────────────────────────────────────────────────────────────────────  
           Class     │            Type             │                                 Rough description                                   
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_1[123X │      [13Xreducible groups[113X       │             stabilisers of totally singular or non-singular subspaces               
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_2[123X │     [13Ximprimitive groups[113X      │    stabilisers of decompositions [23XV=\bigoplus_{i=1}^t V_i, {\rm dim}(V_i)=n/t[123X      
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_3[123X │      [13Xsemilinear groups[113X      │   stabilisers of extension fields of [23X\mathbb{F}_q[123X of prime index dividing [23Xn[123X     
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_4[123X │    [13Xtensor product groups[113X    │         stabilisers of tensor product decompositions [23XV=V_1 \otimes V_2[123X            
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_5[123X │       [13Xsubfield groups[113X       │             stabilisers of subfields of [23X\mathbb{F}_q[123X of prime index               
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_6[123X │   [13Xextraspecial normaliser[113X   │               normalisers of symplectic-type or extraspecial groups                 
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_7[123X │    [13Xtensor induced groups[113X    │ stabilisers of decompositions [23XV=\bigotimes_{i=1}^t V_i, {\rm dim}(V_i)=a, n=a^t[123X   
        ─────────────┼─────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────  
        [23X{\cal C}_8[123X │ [13Xclassical normaliser groups[113X │             groups of similarities of non-degenerate classical forms                
        ─────────────┴─────────────────────────────┴──────────────────────────────────────────────────────────────────────────────────  
  
       [1XTable:[101X Rough descriptions of geometric Aschbacher classes
  
  
