// p5g3r917.m print "\nConstruct Khukhro's 3-generator 5-group counterexample to the Hughes", "\nconjecture, and then repeatedly factor out complements to c^5 in", "\nthe centre of the group, to obtain smaller counterexamples"; d := 3; p := 5; cl := 8; // ngens; prime; class P := pQuotient( FreeGroup(d), p, cl : Exponent := p ); printf "\nB(3,5 : 8) has order %o^%o\n", p, FactoredOrder(P)[1][2]; G := pCoveringGroup(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, 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 "\nNow compute suitable 5th powers of elements outside G'"; load twB35c5; // 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, b and c; and c^5 =", c^p, "\nH is Khukhro's 3-generator anti-Hughes 5-group"; CurrentQ := H; //Repeatedly factor out complements for c^5 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 "\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]; "\ngenerated by a, b and c; and c^5 =", c^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!)";