// p7g2mo.m : This program is an analogue of p7exp.m print "\nConstruct a 2-generator anti-Hughes 7-group of class 14 in", "\nwhich the normal closures of the generators both have class 7.", "\nThen repeatedly factor out complements to [b,a]^7 in the centre", "\nof the group, to obtain smaller counterexamples"; p := 7; P := pQuotientProcess( FreeGroup(2), p, p : Exponent := p ); NextClass( ~P : Exponent := p, MaxOccurrence := [p,p] ); NextClass( ~P : Exponent := p, MaxOccurrence := [p,p] ); NextClass( ~P : Exponent := p, MaxOccurrence := [p,p] ); NextClass( ~P : Exponent := p, MaxOccurrence := [p,p] ); NextClass( ~P : Exponent := p, MaxOccurrence := [p,p] ); NextClass( ~P : Exponent := p, MaxOccurrence := [p,p] ); printf "\nB(2,7 MO77 : 13) has order %o^%o\n", p, FactoredOrder(P)[1][2]; cl14start := FactoredOrder(P)[1][2] + 1; NextClass( ~P : Exponent := 0, MaxOccurrence := [p,p] ); G := ExtractGroup(P); printf "\nThe p-covering group G has order %o^%o and class %o", p, FactoredOrder(G)[1][2], pClass(G); print "\nG is generated by a and b; and [b,a] has order", Order( (b,a) ), "\n[b,a,a] has order", Order( (b,a,a) ), "; [b,a,b] has order", Order( (b,a,b) ), "\ngamma_3(G) is the normal closure of < [b,a,a], [b,a,b] >,", "\nand gamma_3(G) has class at most 4, so gamma_3(G) has exponent 7"; print "\nNow compute suitable 7th powers of elements outside the derived group"; load "twB27c8"; // should be already computed by gettestwords.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 and b; and [b,a] has order", Order( (b,a) ), "\nH is anti-Hughes; the normal closures of a and b both have class 7"; print "\nNow factor out a complement for [b,a]^7 in the multiplier"; ba7 := Eltseq( (b,a)^p ); lastg := FactoredOrder(H)[1][2]; for alive in [cl14start..lastg] do if ba7[alive] eq 0 then continue; end if; // skip if dead // Since the exponent of H.alive is nonzero in [b,a]^7 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 ([cl14start..alive-1] cat [alive+1..lastg]) ] >; printf "CurrentQ with order %o^%o", p, FactoredOrder(CurrentQ)[1][2]; print "\n CurrentQ is generated by a and b; and [b,a]^7 =", (b,a)^p; //Repeatedly factor out complements for [b,a]^7 in the centre Z := Centre(CurrentQ); while Order(Z) ne p do rank := FactoredOrder(Z)[1][2]; printf " Z, its centre, has order %o^%o", p, rank; print "\n Now build a complement for [b,a]^7 in Z"; ZGens := [ CurrentQ!Z.i : i in [1..rank] ]; _, index := Max( Eltseq( (b,a)^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 "\n generated by a and b; and [b,a]^7 =", (b,a)^p; CurrentQ := NextQ; Z := Centre(CurrentQ); end while; printf "The centre is generated by %o and has order %o^%o", CurrentQ!(Z.1), p, FactoredOrder(Z)[1][2]; printf "\nSo this method reduces to an anti-Hughes group with order %o^%o", p, FactoredOrder(CurrentQ)[1][2]; print "\nwhich is as far as we can reduce the group (by this method!)"; end for;