Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys

In [Si], §5. 10D, D. Singmaster asks for the prospective orders of the weather of the Rubik’s dice crew and the way many components of every order there are. (A procedure for choosing it will be defined later during this ebook; see §9. eight. 2. ) this question of Singmaster motivates the subsequent challenge. Ponderable five. nine. 7. (Hard) confirm pG (t) for the Rubik’s dice staff. by means of demanding, I suggest you possibly won’t remedy it in a single day (at least now not except you've gotten entry to a really speedy computer). instance five. nine. 1. For D6 (the dihedral crew of order 12), the producing polynomial is two ∗ t6 + 2 ∗ t3 + 7 ∗ t2 + t; for S8 , the producing polynomial is t + 4t2 + 2t3 + 4t4 + t5 + 5t6 + t7 + t8 + t10 + t12 + t15 ; and for S12 , it really is t + 6t2 + 4t3 + 9t4 + 2t5 + 16t6 + t7 + 4t8 + 2t9 + 6t10 + t11 + 9t12 + 2t14 + 2t15 + t18 + 2t20 + t21 + t24 + t28 + 3t30 + t35 + t42 + t60 . 107 5. nine. CONJUGATION for instance, it follows from the final displayed computaiton that there exists a permutation of order forty two in S12 . In SAGE, those computations are effortless. SAGE sage: sage: sage: sage: tˆ6 + G = DihedralGroup(6) CG = G. conjugacy_classes_representatives() t = var("t") p = sum([tˆ(g. order()) for g in CG]); p tˆ3 + 3*tˆ2 + t looking on your machine, the analogous computation for S8 and S12 may perhaps take decades to finish. Verifying the above examples ‘by hand’ is a difficult workout in itself. SAGE turns out to be useful in those occasions. have a good time fiddling with different teams too! As was once already pointed out, the maximal order (of any point) within the Rubik’s dice crew is 1260. This and the defining equation (5. three) let us know that the measure of the producing polynomial of the Rubik’s dice staff is 1260. Definition five. nine. three. repair a component g in a gaggle G. The set Cl(g) = ClG (g) = {h−1 ∗ g ∗ h | h ∈ G} is named the conjugacy category of g in G. it's the equivalence category of the aspect g less than the relation given by way of conjugation. instance five. nine. 2. this instance will express the best way to use SAGE to compute the entire conjugacy sessions of a bunch and compute the above producing polynomial. SAGE sage: sage: sage: sage: sage: sage: tˆ6 + rubik = CubeGroup() r = rubik. R() u = rubik. U() G = PermutationGroup([rˆ2,uˆ2]) CG = G. conjugacy_classes_representatives() p_G = sum([tˆ(g. order()) for g in CG]); p_G tˆ3 + 3*tˆ2 + t In different phrases, there are 6 non-conjugate parts within the squares subgroup of the dice team. furthermore, pG (t) = t6 + t3 + 3t2 + t. realize that the producing polynomial of this staff is equal to that of the crowd D6 . in truth, those teams are literally isomorphic. 108 5. nine. CONJUGATION notice that if g1 ∈ G is conjugate to g2 ∈ G, then Cl(g1 ) = Cl(g2 ). How do you ‘compute’ conjugacy periods? set of rules enter: a collection S of turbines of a permutation workforce G and an x belonging to G Output: The conjugacy type of x, ClG (x) set of rules classification = {x} for y at school do for g in S do if g*y*gˆ{-1} now not in school then category = category union {g*y*gˆ{-1}} endif endfor endfor be aware that the scale of the checklist classification within the for loop alterations after every one generation of the loop.