  
  [1X8 [33X[0;0YCohomology operations[133X[101X
  
  
  [1X8.1 [33X[0;0YSteenrod operations on the classifying space of a finite [22X2[122X[101X[1X-group[133X[101X
  
  [33X[0;0YThe  following  example  determines  a  presentation for the cohomology ring
  [22XH^∗(Syl_2(M_12),  Z_2)[122X.  The  Lyndon-Hochschild-Serre spectral sequence, and
  Groebner  basis  routines from [12XSingular[112X, are used to determine how much of a
  resolution to compute for the presentation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SylowSubgroup(MathieuGroup(12),2);;[127X[104X
    [4X[25Xgap>[125X [27XMod2CohomologyRingPresentation(G);[127X[104X
    [4X[28XGraded algebra GF(2)[ x_1, x_2, x_3, x_4, x_5, x_6, x_7 ] / [128X[104X
    [4X[28X[ x_2*x_3, x_1*x_2, x_2*x_4, x_3^3+x_3*x_5, [128X[104X
    [4X[28X  x_1^2*x_4+x_1*x_3*x_4+x_3^2*x_4+x_3^2*x_5+x_1*x_6+x_4^2+x_4*x_5, [128X[104X
    [4X[28X  x_1^2*x_3^2+x_1*x_3*x_5+x_3^2*x_5+x_3*x_6, [128X[104X
    [4X[28X  x_1^3*x_3+x_3^2*x_4+x_3^2*x_5+x_1*x_6+x_3*x_6+x_4*x_5, [128X[104X
    [4X[28X  x_1*x_3^2*x_4+x_1*x_3*x_6+x_1*x_4*x_5+x_3*x_4^2+x_3*x_4*x_5+x_3*x_5^\[128X[104X
    [4X[28X2+x_4*x_6, x_1^2*x_3*x_5+x_1*x_3^2*x_5+x_3^2*x_6+x_3*x_5^2, [128X[104X
    [4X[28X  x_3^2*x_4^2+x_3^2*x_5^2+x_1*x_5*x_6+x_3*x_4*x_6+x_4*x_5^2, [128X[104X
    [4X[28X  x_1*x_3*x_4^2+x_1*x_3*x_4*x_5+x_1*x_3*x_5^2+x_3^2*x_5^2+x_1*x_4*x_6+\[128X[104X
    [4X[28Xx_2^2*x_7+x_2*x_5*x_6+x_3*x_4*x_6+x_3*x_5*x_6+x_4^2*x_5+x_4*x_5^2+x_6^\[128X[104X
    [4X[28X2, x_1*x_3^2*x_6+x_3^2*x_4*x_5+x_1*x_5*x_6+x_4*x_5^2, [128X[104X
    [4X[28X  x_1^2*x_3*x_6+x_1*x_5*x_6+x_2^2*x_7+x_2*x_5*x_6+x_3*x_5*x_6+x_6^2 [128X[104X
    [4X[28X ] with indeterminate degrees [ 1, 1, 1, 2, 2, 3, 4 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  command  [10XCohomologicalData(G,n)[110X  prints  complete  information  for the
  cohomology  ring  [22XH^∗(G, Z_2 )[122X of a [22X2[122X-group [22XG[122X provided that the integer [22Xn[122X is
  at  least  the  maximal degree of a relator in a minimal set of relators for
  the  ring.  Groebner basis routines from [12XSingular[112X are called involved in the
  example.[133X
  
  [33X[0;0YThe  following example produces complete information on the Steenrod algebra
  of group number [22X8[122X in [12XGAP[112X's library of groups of order [22X32[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XGroup number: 8[128X[104X
    [4X[28XGroup description: C2 . ((C4 x C2) : C2) = (C2 x C2) . (C4 x C2)[128X[104X
    [4X[28X[128X[104X
    [4X[28XCohomology generators[128X[104X
    [4X[28XDegree 1: a, b[128X[104X
    [4X[28XDegree 2: c, d[128X[104X
    [4X[28XDegree 3: e[128X[104X
    [4X[28XDegree 5: f, g[128X[104X
    [4X[28XDegree 6: h[128X[104X
    [4X[28XDegree 8: p[128X[104X
    [4X[28X[128X[104X
    [4X[28XCohomology relations[128X[104X
    [4X[28X1: f^2[128X[104X
    [4X[28X2: c*h+e*f[128X[104X
    [4X[28X3: c*f[128X[104X
    [4X[28X4: b*h+c*g[128X[104X
    [4X[28X5: b*e+c*d[128X[104X
    [4X[28X6: a*h[128X[104X
    [4X[28X7: a*g[128X[104X
    [4X[28X8: a*f+b*f[128X[104X
    [4X[28X9: a*e+c^2[128X[104X
    [4X[28X10: a*c[128X[104X
    [4X[28X11: a*b[128X[104X
    [4X[28X12: a^2[128X[104X
    [4X[28X13: d*e*h+e^2*g+f*h[128X[104X
    [4X[28X14: d^2*h+d*e*f+d*e*g+f*g[128X[104X
    [4X[28X15: c^2*d+b*f[128X[104X
    [4X[28X16: b*c*g+e*f[128X[104X
    [4X[28X17: b*c*d+c*e[128X[104X
    [4X[28X18: b^2*g+d*f[128X[104X
    [4X[28X19: b^2*c+c^2[128X[104X
    [4X[28X20: b^3+a*d[128X[104X
    [4X[28X21: c*d^2*e+c*d*g+d^2*f+e*h[128X[104X
    [4X[28X22: c*d^3+d*e^2+d*h+e*f+e*g[128X[104X
    [4X[28X23: b^2*d^2+c*d^2+b*f+e^2[128X[104X
    [4X[28X24: b^3*d[128X[104X
    [4X[28X25: d^3*e^2+d^2*e*f+c^2*p+h^2[128X[104X
    [4X[28X26: d^4*e+b*c*p+e^2*g+g*h[128X[104X
    [4X[28X27: d^5+b*d^2*g+b^2*p+f*g+g^2[128X[104X
    [4X[28X[128X[104X
    [4X[28XPoincare series[128X[104X
    [4X[28X(x^5+x^2+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1)[128X[104X
    [4X[28X[128X[104X
    [4X[28XSteenrod squares[128X[104X
    [4X[28XSq^1(c)=0[128X[104X
    [4X[28XSq^1(d)=b*b*b+d*b[128X[104X
    [4X[28XSq^1(e)=c*b*b[128X[104X
    [4X[28XSq^2(e)=e*d+f[128X[104X
    [4X[28XSq^1(f)=c*d*b*b+d*d*b*b[128X[104X
    [4X[28XSq^2(f)=g*b*b[128X[104X
    [4X[28XSq^4(f)=p*a[128X[104X
    [4X[28XSq^1(g)=d*d*d+g*b[128X[104X
    [4X[28XSq^2(g)=0[128X[104X
    [4X[28XSq^4(g)=c*d*d*d*b+g*d*b*b+g*d*d+p*a+p*b[128X[104X
    [4X[28XSq^1(h)=c*d*d*b+e*d*d[128X[104X
    [4X[28XSq^2(h)=d*d*d*b*b+c*d*d*d+g*c*b[128X[104X
    [4X[28XSq^4(h)=d*d*d*d*b*b+g*e*d+p*c[128X[104X
    [4X[28XSq^1(p)=c*d*d*d*b[128X[104X
    [4X[28XSq^2(p)=d*d*d*d*b*b+c*d*d*d*d[128X[104X
    [4X[28XSq^4(p)=d*d*d*d*d*b*b+d*d*d*d*d*d+g*d*d*d*b+g*g*d+p*d*d[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X8.2 [33X[0;0YSteenrod operations on the classifying space of a finite [22Xp[122X[101X[1X-group[133X[101X
  
  [33X[0;0YThe  following  example  constructs  the  first  eight  degrees of the mod-[22X3[122X
  cohomology  ring  [22XH^∗(G,  Z_3)[122X  for the group [22XG[122X number 4 in [12XGAP[112X's library of
  groups  of order [22X81[122X. It determines a minimal set of ring generators lying in
  degree  [22Xle  8[122X  and  it evaluates the Bockstein operator on these generators.
  Steenrod  powers  for  [22Xpge  3[122X  are not implemented as no efficient method of
  implementation is known.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(81,4);;[127X[104X
    [4X[25Xgap>[125X [27XA:=ModPSteenrodAlgebra(G,8);;[127X[104X
    [4X[25Xgap>[125X [27XList(ModPRingGenerators(A),x->Bockstein(A,x));[127X[104X
    [4X[28X[ 0*v.1, 0*v.1, v.5, 0*v.1, (Z(3))*v.7+v.8+(Z(3))*v.9 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
