// wallcl9.m print "\nConstruct the largest class 9 quotient of Wall's 3-generator", "\nanti-Hughes 5-group.", "\nThen repeatedly factor out complements to c^5 in the centre", "\nof the group, to obtain smaller counterexamples"; p := 5; F := FreeGroup(3); Q := quo < F | (b,a,a,a,b,a,b,a), (b,a,a,a,b,a,b,b), (b,a,a,a,b,b,a,b), (b,a,a,b,b,a,b,b) >; P := pQuotientProcess( Q, p, 2 : Exponent := p ); NextClass( ~P : Exponent := p, MaxOccurrence := [0,0,1] ); NextClass( ~P : Exponent := p, MaxOccurrence := [0,0,1] ); NextClass( ~P : Exponent := p, MaxOccurrence := [0,0,1] ); NextClass( ~P : Exponent := p, MaxOccurrence := [0,0,1] ); NextClass( ~P : Exponent := p, MaxOccurrence := [0,0,1] ); NextClass( ~P : Exponent := p, MaxOccurrence := [0,0,1] ); // Note: this implementation of the p-quotient algorithm uses // the lower exponent-\$p\$-central series so we must allow // 2 (not 1) occurrences of c in the covering group // to allow the order of c to become 25 in the cover NextClass( ~P : Exponent := 0, MaxOccurrence := [0,0,2] ); G := ExtractGroup(P); printf "\nThe p-covering group G has order %o^%o and class %o", FactoredOrder(G)[1][1], FactoredOrder(G)[1][2], pClass(G); print "\nG is generated by a, b and c; and c has order", Order(c), "\n[b,a] has order", Order( (b,a) ), "; [c,a] has order", Order( (c,a) ), "; [c,b] has order", Order( (c,b) ), "\ngamma_2(G) is the normal closure of < [b,a], [c,a], [c,b] >,", "\nand gamma_2(G) has class at most 4, so gamma_2(G) has exponent 5"; print "Also factor out the derived group of the normal closure of c", "\n(This became non-trivial in the p-cover computation)"; C := ncl< G | c >; C1 := DerivedGroup(C); G := quo< G | C1 >; printf "The group now has order %o^%o", p, FactoredOrder(G)[1][2]; print "\nNow compute suitable 5th powers of elements outside G'"; load twB35c5mo441; // should be already computed by gettestwq.m S := [ x^p : x in testwords ]; H := quo< G | S >; printf "The quotient group H has order %o^%o", p, FactoredOrder(H)[1][2]; print "\nH is generated by a, b and c; and c^5 =", c^p, "\nH is anti-Hughes; the normal closure of c is abelian", "\nand this implies that gamma_3(G) has class 3", "\nso every element cg, g in gamma_3, has order 25", "\nWe also have:", "c[b,a] has order", Order(c*(b,a)), "; c[c,a] has order", Order(c*(c,a)), "; c[c,b] has order", Order(c*(c,b)), "\nSo the Hughes subgroup is G'"; "\nNow factor out a complement for c^5 in the multiplier"; cl9start := pRanks(H)[8] + 1; lastg := FactoredOrder(H)[1][2]; alive := lastg; // Since the exponent of H.alive is nonzero in c^5 we // choose a complement of H.alive print "\nFactoring out a complement of", H.alive, "gives us a smaller anti-Hughes group"; CurrentQ := quo< H | [ H.i : i in [cl9start..alive-1] ] >; printf "CurrentQ with order %o^%o", FactoredOrder(CurrentQ)[1][1], FactoredOrder(CurrentQ)[1][2]; print "\nCurrentQ is generated by a, b and c; and c^5 =", c^p, "\n"; Z := Center(CurrentQ); while Order(Z) ne p do rank := FactoredOrder(Z)[1][2]; printf "Z, its centre, has order %o^%o", p, rank; print "\nNow build a complement for c^5 in Z\n"; ZGens := [ CurrentQ!Z.i : i in [1..rank] ]; _, index := Max( Eltseq( c^p ) ); ComplGens := [ ZGens[i] * (CurrentQ.index)^-Eltseq(ZGens[i])[index] : i in [1..rank] ]; NextQ := quo< CurrentQ | ComplGens >; printf "Factor it out to get an anti-Hughes group of order %o^%o", p, FactoredOrder(NextQ)[1][2]; print "\ngenerated by a, b and c; and c^5 =", c^p; CurrentQ := NextQ; Z := Center(CurrentQ); end while; printf "The centre is generated by %o and has order %o^%o", CurrentQ!(Z.1), FactoredOrder(Z)[1][1], FactoredOrder(Z)[1][2]; printf "\nSo this method reduces to an anti-Hughes group with order %o^%o", FactoredOrder(CurrentQ)[1][1], FactoredOrder(CurrentQ)[1][2]; print "\nwhich is as far as we can reduce the group (by this method!)";