Slide-and-swap permutation groups. Onyebuchi Ekenta, Han Gil Jang and Jacob A. Siehler. (Communicated by Joseph A. Gallian)

Transcription

1 msp INVOLVE 7:1 (2014) x.oi.or/ /involv Sli-n-swp prmuttion roups Onyui Eknt, Hn Gil Jn n Jo A. Silr (Communit y Josp A. Gllin) W prsnt simpl til-sliin m tt n ply on ny 3-rulr rp, nrtin prmuttion roup on t vrtis. W lssiy t rsultin prmuttion roups n otin novl prsnttion or t simpl roup o 168 lmnts. From sliin tils to simpl roups T sliin tils o t notorious itn puzzl (rrn in 4 4 rry wit on squr missin) r n ojt lsson in prity: Wi prmuttions o t numr tils n iv? Prisly t vn prmuttions. Put notr wy, t movs o t itn puzzl nrt t ltrntin roup A 15. Aron Arr [1999] ivs us tiy proo o tis olklori t. R. M. Wilson [1974] onsirs til-sliin ms on ritrry rps s nrliztion o t itn puzzl, n lssiis t prmuttion roups wi n nrt y ts ms. Brily, t prmuttion roup or til-sliin m on rp wit k vrtis is nrlly itr t ltrntin roup A k 1 i t rp is iprtit or t ull symmtri roup S k 1 i it is not. Tr is only on intrstin xption, 7-vrtx rp wi nrts roup o just 120 prmuttions. Wilson prsnts tis roup s PGL(2, F 5 ), t roup o Möius trnsormtions ovr t il o iv lmnts, ut it is isomorpi to t symmtri roup S 5. Fink n Guy [2009] iv torou isussion o tis intrstin, xptionl s, wi ty rr to s t triky six puzzl. Jon Conwy [1997; 2006] uss til-sliin m on 13-point projtiv pln to nrt t Mtiu roup M 12. Tis m is u M 13. In M 13, movin til to t opn point lso rquirs swppin two otr tils t t sm tim. Howvr, t ruls o t m r spii to t projtiv pln on wi it is ply, n it os not nrliz in ny ovious wy to mily o ms on lrr projtiv plns or otr omintoril struturs. MSC2010: 20B15, 91A43. Kywors: simpl roup, prmuttion roup, primitiv roup, rp tory. 41

2 42 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER Hr, w onsir prmuttion m wi n ply on ny 3-rulr rp, usin sli-n-swp rul inspir y Conwy s M 13. W lssiy t rsultin prmuttion roups n otin nrl rsult similr to Wilson s torm, wit just on intrstin xptionl s. T xption ivs novl n lmntry prsnttion or t simpl roup o 168 lmnts. Lik Ruik s u n t otr prmuttion ms w v mntion, ours n trt purly s puzzl, wr you srml pis y movin tm out n tn try to rturn tm to tir initil oniurtion or, prps, try to iv som otr ol oniurtion tt s n pos s prolm. W v m plyl vrsion o t m [Silr 2011] s n i to unrstnin t ruls. In tis rtil, w o not iv ny loritms or unsrmlin t pis on ivn rp, so solvin t puzzl in tt sns soul rmin n njoyl lln i you r so inlin. Ruls o t sli-n-swp m Lt 0 3-rulr rp (wi w my s wll ssum to onnt) wit ll vrtis. T m on 0 ins wit on vrtx unovr n o t rminin vrtis ovr y til wit t sm ll. At ny st in t m, you mk mov s ollows: Coos vrtx v jnt to t urrnt unovr vrtx n sli t til on v into t unovr position (unovrin v in t pross); n, t t sm tim, swp t tils on t otr two niors o v. S t irst two irms in Fiur 1, rprsntin sinl mov wr til slis rom vrtx into t unovr vrtx n t tils on t otr two niors o vrtx r swpp. W not t mov o sliin til rom vrtx to vrtx y [, ]. I tis sms ountrintuitiv, tink o t ol itsl s spil lnk til wi movs lon t vrtis in t orr tt ty r list wn t mov is ply. Lonr mov squns r xprss similrly: [,,,...] is t mov squn wi strts wit t ol on vrtx, movs it to vrtx, rom tr to vrtx, n so on. Fiur 1 sows squn o our movs ply on t 8-vrtx u rp. T rsultin prmuttion o tils n xprss in yl orm s ( ) wn tis squn is ply, t til wi ins on vrtx movs to vrtx ; t til on vrtx movs to vrtx ; n so on. I P =,,,... is pt in 0, tn plyin P mns plyin t mov squn [,,,...]. In trms o til movmnts, tt mns irst movin t til on vrtx to t vnt vrtx, tn movin t til on vrtx to t nwly vt vrtx, n so on, wit t ompnyin swps t stp. T ol itsl pros rom to to n so on, in t orr ty r list. Unlik t itn puzzl or M 13, t srmlin tt ppns s rsult o

3 SLIDE-AND-SWAP PERMUTATION GROUPS Fiur 1. [,,,, ]=( ) on t 8-vrtx u rp. plyin pt P in tis m nnot unon simply y plyin its rvrs (mnin t sm pt s P, just trvrs in t opposit irtion). Howvr, w o v si rsult out invrtiility: Proposition 1. Any ll mov squn in t sli-n-swp m n unon (rturnin t tils to tir position or t squn ws ply) y notr ll mov squn. Proo. Suppos vrtx x 0 is initilly mpty n w prmut t tils y plyin P =[x 0,,...,x n ]. Lt R =[x n, x n 1,...,x 0 ]. T rsult o P ollow y R is prmuttion (o init orr, sin tr r only initly mny tils!) wi rturns t ol to x 0. From tt point w n ply P ollow y R rptly until ll t tils (n t ol) v rturn to tir initil position. You my lso not tt sinl sli [x 0, ] n unon y [, x 0,, x 0 ]. Consquntly, lonr mov squn [x 0,,...,x n ] n unon on stp t tim, s ollows: First uno t inl sli y plyin [x n, x n 1, x n, x n 1 ], tn uno t on or tt y [x n 1, x n 2, x n 1, x n 2 ], n so on. Sli-n-swp loop roups Now, suppos w in m on onnt, 3-rulr rp 0 wit k vrtis, vrtx initilly unovr. T prmuttions wi rturn t ol to its initil vrtx (s in Fiur 1) orm roup unr omposition. W ll tis t loop roup or 0 s t n not it &. Sin t si movs in t m r ll oul trnspositions, & is lwys suroup o t ltrntin roup A k 1. T nottion

4 44 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER susts tt t loop roup pns on t oi o t initil unovr vrtx, ut up to isomorpism t oi os not mttr. Proposition 2. For ny vrtis n in 0, t roups & n & r isomorpi. Proo. Sin 0 is onnt, oos pt P rom to n lt t prmuttion inu y plyin P. T mppin 7! 1 ins omomorpism rom & to &. Tis omomorpism s 7! 1 s its invrs, so t two roups r isomorpi (in, ty r onjut insi t symmtri roup S k ). For tis rson, w will nort omit ny rrn to t unovr vrtx n rr to t loop roup ssoit to rp. Proposition 3. T loop roup o t ttrron (tt is, t omplt rp on 4 vrtis) is trivil. T proo o tis is lt s n xris. Lrr rps nrt nontrivil roups, owvr, n nturl lri prolm is to trmin, up to isomorpism, wi prmuttion roups n rliz s sli-n-swp loop roups. Tt prolm is ntirly rsolv y t ollowin torms, wi w will prov in t susqunt stions. Nottion. From now on, 0 will lwys not onnt, 3-rulr rp on k vrtis, n & will not its loop roup (wit t unrstnin tt t oi o mpty vrtx osn t mttr). Torm 1. I 0 is not t u or ttrron, tn & is isomorpi to t ltrntin roup A k 1. Torm 2. T loop roup o t u is isomorpi to GL(3, F 2 ), t simpl roup o 168 lmnts. T rsmln to Wilson s rsults or orinry til-sliin ms on rps sms unnny. Funmntl trms n propositions Dixon s prolm ook [1973] is ny rrn or t lmntry tory o prmuttion roups, n t mtril is vlop in pt in [Dixon n Mortimr 1996]. Hr, w n only w si initions n proprtis. Lt G roup o prmuttions on st X. T orit o n lmnt x 2 X is { (x) 2 G}. Ts orits orm prtition o X. I tr is only on orit (wi ontins ll t lmnts o X), tn G is si to trnsitiv. A nonmpty st B X is ll lok or G i or vry 2 G, itr (B) = B or (B) \ B =?. T st X itsl is vintly lok, s r ll sinlton susts o X; ts r ll trivil loks. G is si to primitiv i ll loks or G r trivil; otrwis, i nontrivil loks xist, G is imprimitiv.

5 SLIDE-AND-SWAP PERMUTATION GROUPS 45 I G is trnsitiv n B is ny lok, tn t sts (B), wr 2 G, prtition X into isjoint, nonmpty sts, o wi is lok. Su prtition o X is ll systm o imprimitivity or G. W onsir & to roup o prmuttions on t nonmpty vrtis o 0. T ollowin proposition is lss ovious or sli-n-swp ms tn it is or orinry til-sliin ms. Som tim spnt wit plyl vrsion o t m [Silr 2011], tryin to mov ivn til to ivn vrtx, my lpul in unrstnin t iiultis. Proposition 4. I 0 is not t ttrron, tn & is trnsitiv. Lik Proposition 2, tis is usul t to rliz rom t outst. It ollows rom t proo o Proposition 9, owvr, so w o not inlu sprt proo t tis point. Our proo o Torm 1 pns on t ollowin nrl rsult: Proposition 5 [Wilson 1974]. Lt G trnsitiv prmuttion roup on st X n suppos tt G ontins 3-yl. I G is primitiv, tn G ontins t ltrntin roup on X. Gnrtin t ltrntin roup T nxt w rsults sow tt, in nrl, our loop roups stisy t ypotss o Proposition 5. First, w stlis t prsn o 3-yls. Proposition 6. I 0 is not t ttrron or u, tn & ontins 3-yls. Proo. First, not tt t isomorpism in Proposition 2 is rliz y onjution witin t symmtri roup, wi prsrvs yl typs. For tis rson, i t roup s t ny vrtx ontins 3-yl, tn tis is tru t vry vrtx, n so w n oos t mpty vrtx t our onvnin. In t ollowin, t lls (, x 2, n so on) nm t vrtis o t rp. It osn t mttr wi tils r on wi vrtis, xpt tt t vrtx ll x 0 is t initilly opn vrtx. W onsir tr ss. Cs 1. I 0 ontins trinl {x 0,, x 2 }, tn (sin 0 is not ttrron) two vrtis in t trinl must v istint niors. Suppos x 0 n v niors x 3 n x 4, lik tis: x 4 x 3 x 0 x 2 In tis s, [x 0,, x 0 ] = (x 2 x 4 x 3 ).

6 46 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER Cs 2. Suppos 0 os not ontin trinl, ut ontins pt x 0,, x 2 wr x 0 n x 2 v no niors in ommon xpt : x 4 x 5 x 0 x 2 x 3 x 7 x 6 Tn [x 0,, x 2,, x 0 ] 2 = ( x 2 x 7 ). Similrly, i 0 s no trinl ut s pt x 0,, x 2 wr x 0 n x 2 v ll tr niors in ommon (sy, x 4 = x 5 n x 3 =x 6 in t iur), tn [x 0,, x 2,, x 0 ] = ( x 7 x 2 ). Cs 3. T only rminin s to onsir is rp 0 wit no trinls, in wi t npoints o vry pt o lnt two v xtly two niors in ommon. In An tls o rps, R n Wilson [1998] sow tt tr r only six 3-rulr rps o imtr lss tn tr, inluin t ttrron, n non o tm stisy ts ypotss. Ts r ll smll rps n t lim is sy to vriy y insption. Tror w my ssum tt t imtr o 0 is t lst 3. W lim tt in tis s 0 n only t u. Bin wit pt,,,, wr t istn rom to is 3 (so tr is no sortr pt rom to ). Lt v t otr ommon nior o n, n w t otr ommon nior o n : v Now t pt,,w implis tt n w v notr ommon nior x. Similrly t pt,, vimplis tt n v must v notr ommon nior y. Tis rins us to t ollowin sitution: w v y x w Sin t rp is 3-rulr, x ns notr. I x wr onnt to som otr vrtx z not lry sown, tn t pt z, x, woul imply tt tr is notr vrtx jnt to ot n z. Tis woul imply tt itr, or on o t vrtis jnt to in t prin iur, s n itionl, wi is

7 SLIDE-AND-SWAP PERMUTATION GROUPS 47 impossil; o tos vrtis lry s tr s. It ollows tt x n y r jnt. At tis point ll t vrtis v ll tr s ount or, n t rsultin rp is t u, s lim. Proposition 7. Suppos tt or vry pir o jnt vrtis x n y in 0 tr is p-yl 2 & wr p is prim n (x) = y. Tn & is primitiv. Proo. Suppos t nonmpty vrtis r prtition into loks B 1, B 2,...,B n ormin systm o imprimitivity or &. Lt B i n B j two loks wi ontin jnt vrtis w i n w j, rsptivly. By ypotsis, w my oos p-yl 2 & wit (w i ) = w j. Consir s prmuttion o loks, ts nontrivilly (snin B i to B j ), n sin p is prim, tis implis tt ts wit orr p. Howvr, only movs p vrtis, so tr n only p loks involv n only on vrtx in o tos loks, inluin B i. Tis rumnt n ppli wit ny lok plyin t rol o B i n ny jnt lok plyin t rol o B j. Tus, sin 0 is onnt, itr ll loks r sinltons or tr is only sinl lok. Sin no nontrivil loks r possil, & is primitiv. Proposition 8. Givn ny two jnt, nonmpty vrtis x n y in 0, tr is mov squn wi lvs t tils on x n y ix wil positionin t ol jnt to on o tos two vrtis. Proo. I t mpty vrtx is lry jnt to x or y, no movs r n. Otrwis, lt Q t sortst possil pt innin t t mpty vrtx, wit t proprty tt t nin vrtx o Q s istn o two rom x or istn o two rom y. No point on Q n jnt to itr x or y, sin t prvious point woul v istn o two, n sortr pt oul onstrut. It ollows tt t prmuttion inu y plyin Q lvs t tils on x n y ix. So, ssum tt w v ply Q, nin wit t ol on vrtx x 0 wi is istn two rom on o t ivn vrtis witout loss o nrlity, sy it s istn two rom y. T ol now is to mov t ol onto vrtx wi is jnt to y, still witout isturin t tils on x n y. Tr r tr lol oniurtions to onsir, n t propr movs to pro in s r sown in Fiurs 2 4. x y x 0 Fiur 2. [x 0,, x 0, ]=(x 0 )( ).

8 48 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER x y x 0 Fiur 3. [x 0,, x 0,, x 0, ]=(x 0 )( ). x y x 0 Fiur 4. [x 0,, x 0, ]=(x 0 )( ). In s, w t t sir rsult. Vrtx (jnt to y) is vt, wil t tils on x n y rmin ix. W justiy t lim tt ts tr r t only ss s ollows: Sin x 0 is not jnt to itr x or y it must v two niors otr tn x, y or. An must v on itionl nior otr tn y or x 0. Tis itionl nior my x, or on o t niors o x 0, or point istint rom ot x n t niors o x 0, prisly t tr ss w v onsir. Proposition 9. I 0 is not t ttrron, tn t ypotss o Proposition 7 r stisi, n & is primitiv. Proo. On in, lt x n y ny two jnt, nonmpty vrtis in 0. Assum t mpty vrtx is jnt to y. Now, t possil oniurtions o t rp nr x n y r summriz in t ollowin iur, wr x 0 is t mpty vrtx n is vrtx jnt to y otr tn x n x 0. E ott my or my not xist. I ll s xist simultnously tn 0 is t ttrron. Tt lvs svn nontrivil ss to onsir. Tr r six ss in wi t lst on ott xists; ty r sown in

9 SLIDE-AND-SWAP PERMUTATION GROUPS 49 Fiur 5. An x 0 -. [x 0,, y, x 0 ] inus itr (x y ) or (x y ). Fiur 6. An x-x 0. [x 0, x, y, x 0 ] inus itr (x y) or (x y). Fiur 7. An x-. [x 0, y, x 0, y,, y, x 0 ] inus itr (x y) or (x y). Fiur 8. x- n x 0 - s. [x 0,, x 0 ]=(x y). Fiurs In s w xiit pt wi n ply to nrt yl o prim lnt (itr 3- or 5-yl) snin x to y. Not tt i ott is omitt, t vrtis tt onnts must v n to som otr vrtx not pprin in t iur t t ottom o prvious p. Ts xtr vrtis r not nssrily istint, so tr r som suss to onsir. In our iurs, vrtis not on t pt or jnt to point on t pt r not

10 50 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER Fiur 9. x- n x-x 0 s. In t irst s, [x 0, y,, y, x 0 ]= (x y). In t son s, [x 0, y, x 0, y, x 0, y,, y, x 0 ]= (x y). Fiur 10. x- n x 0 - s. [x 0,, x 0 ]=(x y). u v Fiur 11. No s mon {x, x 0, }. [x 0, y,, y, x 0 ] inus itr ( )( )(x y) (wi n squr to t t sir 3-yl) or (x y). rwn us ty v no t on t rsultin prmuttion. Fiur 11 sows t s wr non o t ott s r prsnt, n n x 0 sr 0 or 2 niors otr tn y. Tt lvs t s wr non o t ott s r prsnt, n n x 0 v xtly on ommon nior u. T tir nior o u is itr on o t points on t rp otr tn y, or point o t rp. So tr r totl o our suss to l wit in tis s n ty r sown in Fiurs In vry s, w prou yl o prim lnt wi sns x to y. But w n wit t provisionl ssumption tt t initilly mpty vrtx is jnt to y. In nrl, owvr, Proposition 8 n ppli to prou prmuttion wi movs t mpty vrtx jnt to y wil lvin x n y ix. Conjutin y ivs t sir yl in &. Rmrk. Sin ny vrtx my mov to ny jnt vrtx y mns o ts yls, & is trnsitiv, s w ssrt in Proposition 4.

11 SLIDE-AND-SWAP PERMUTATION GROUPS 51 u Fiur 12. [x 0, u,, y, x 0 ]=( uxy). u Fiur 13. [x 0, u,, y, x 0 ]=(x y u). u Fiur 14. [x 0, y, x 0, u,, u, x 0 ]=(x uy). u q Fiur 15. [x 0, u,, y, x 0 ]=( uxy q). Proo o Torm 1. Our min rsult now ollows quikly. I 0 is not t u or ttrron, tn Proposition 6 sows tt & ontins 3-yls. Proposition 9 sows tt & is trnsitiv n primitiv. Tus Proposition 5 pplis to & n w onlu tt & ontins ll vn prmuttions o t nonmpty vrtis. T xptionl u Now, w nlyz t loop roup o t u. Initilly, omputr lultion rvl tt tis roup s only 168 lmnts (inst o t xpt 7!/2 = 2520 in A 7 ). T numr 168 is milir to lrists s t orr o GL(3, F 2 ), t roup

12 52 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER Fiur 16. Vrtis o t u ll wit vtors o F 3 2. q Fiur 17. Vrtis o t u ll wit unknown vtors. o invrtil 3 3 mtris ovr t il o two lmnts, n t son-smllst nonlin simpl roup. To stlis onntion twn tis roup n t u, w ll t vrtis wit 3-imnsionl vtors ovr F 2, s in Fiur 16. For rvity, w writ vtor 1, 2, 3 i s 3-it inry strin Wit su llin, movs in t m n intrprt s prmuttions o F 3 2. T prtiulr llin in Fiur 16 s t proprty tt t sum (mo 2, o ours) o ny til totr wit its tr jnt tils is zro, n w will ll ny rrnmnt o vtor tils wit tis proprty lolly zro rrnmnt. Proposition 10. T prmuttion o F 3 2 inu y sinl mov on u wit lolly zro rrnmnt o vtors is n in trnsormtion. Proo. Lt,..., n q t it vtors o F 3 2, llin t vrtis o t u in lolly zro rrnmnt s in Fiur 17. I w in = + q, = + q, = + q, t ollowin itionl rltions ollow quikly rom t lolly zro onition: + = + q, + = + q, + = + q, + + = + q. Not tt ll t linr omintions o,, n r istint, so {,, } is linrly inpnnt st (n, in t, sis or F 3 2 ).

13 SLIDE-AND-SWAP PERMUTATION GROUPS 53 Now, onsir sli-n-swp mov. By symmtry w n ssum tt q is mpty n w sli til into t ol rom. Tis inus prmuttion ' wit '(q) =, '() = q, '() =, '() =, n t otr points rminin ix. Din ˆ' y ˆ'(x) = '(x + q) +. Applyin ˆ' to our sis lmnts ivs ˆ'( ) = + +, ˆ'( ) =, ˆ'( ) =, rom wi w n vriy t r wy (tt is, y kin vry linr omintion o sis lmnts) tt ˆ' is linr: ˆ'( + ) =ˆ'( + q) = + = + =ˆ'( ) +ˆ'( ), ˆ'( + )= ˆ'( + q) = + = + =ˆ'( ) +ˆ'( ), ˆ'( + )= ˆ'( + q) = q + = + =ˆ'( ) +ˆ'( ), ˆ'( + + )= ˆ'( + q) = + = =ˆ'( ) +ˆ'( ) +ˆ'( ), n o ours ˆ'(0) = '(q)+ = + = 0. Tus, wit rspt to t sis {,, }, ˆ' is rprsnt y t mtrix M 1 1 0A, n ' is sri y t ormul '(x) = M(x +q)+, or '(x) = Mx+(Mq+), n in trnsormtion s lim. Proo o Torm 2. Bin t m wit vrtis ll y vtors in lolly zro rrnmnt n t 000 vrtx opn. By t prin proposition, ny squn o slis is omposition o in trnsormtions (wi is in n in trnsormtion). T lotion o t ol lwys rvls t trnsltion prt o t trnsormtion, so t loop roup (orrsponin to slis wr t ol rturns to 000) onsists o linr trnsormtions n is ontin in GL(3, F 2 ). To omplt t proo, w simply xiit w lmnts o t roup. Rturnin to t u in Fiur 17 n supposin q is initilly opn, onsir t lmnts [q,, q,,,,,, q]=( )( ), [q,, q]=( )( ), wi nrt irl roup o it lmnts. Also, [q,,,, q]=( )( ), [q,,,, q]=( ). Suroups o orr 8, 3, n 7 imply roup o orr t lst 168, n so t loop roup o t u is not just ontin in, ut qul to GL(3, F 2 ).

14 54 ONYEBUCHI EKENTA, HAN GIL JANG AND JACOB A. SIEHLER Eiint solutions n otr qustions In prinipl, t mto o proo tt w us to lssiy t loop roups oul likly turn into n loritm or solvin srml puzzl on ny ivn rp, sin w sow ow to prou smll, loliz yls t ny point on t rp, wi oul us to mirt pis to tir pproprit lotions. In prti, tis sort o solution tks mny mor movs tn t optiml solution. In orinry til-sliin ms, t prolm o inin n optiml solution or srml stt is NP-omplt [Golri 1984; Rtnr n Wrmut 1990]. W o not know i t sm is tru or sli-n-swp ms; tis is qustion or utur ppr. T sli-n-swp vrint is lso unusul in tt t numr o movs rquir to iv position rom strt my irnt rom t numr o movs rquir to rturn it to strt. Tis spt o t puzzl s no ountrprt in otr sliin or twistin prmuttion puzzls tt w r milir wit, n t rltionsip twn istn to n istn rom strt is worty o som urtr nlysis. Conwy, Elkis n Mrtin [Conwy t l. 2006] v sown ow to us ulity o t projtiv pln to prou n outr utomorpism o t Mtiu roup M 12. It woul intrstin i t symmtris o t u n t sli-n-swp m ruls llow similr onstrution o outr utomorpisms or GL(3, F 2 ), ut w v not yt isovr ow to o tis, n it rmins sujt or urtr invstition. Rrns [Arr 1999] A. F. Arr, A morn trtmnt o t 15 puzzl, Amr. Mt. Montly 106:9 (1999), MR 2001:05004 Zl [Conwy 1997] J. H. Conwy, M 13, pp in Survys in omintoris, 1997, it y R. A. Bily, Lonon Mt. So. Ltur Not Sr. 241, Cmri Univ. Prss, MR 98i:20003 Zl [Conwy t l. 2006] J. H. Conwy, N. D. Elkis, n J. L. Mrtin, T Mtiu roup M 12 n its psuoroup xtnsion M 13, Exprimnt. Mt. 15:2 (2006), MR 2007k:20005 Zl [Dixon 1973] J. D. Dixon, Prolms in roup tory, Dovr, [Dixon n Mortimr 1996] J. D. Dixon n B. Mortimr, Prmuttion roups, Grut Txts in Mtmtis 163, Sprinr, Nw York, MR 98m:20003 Zl [Fink n Guy 2009] A. Fink n R. Guy, Rik s triky six puzzl: S 5 sits spilly in S 6, Mt. M. 82:2 (2009), MR Zl [Golri 1984] O. Golri, Finin t sortst mov-squn in t rp-nrliz 15- puzzl is NP-r, 1984, Avill t ttp://tinyurl/15puzz-p. prprint. [Rtnr n Wrmut 1990] D. Rtnr n M. Wrmut, T (n 2 1)-puzzl n rlt rlotion prolms, J. Symoli Comput. 10:2 (1990), MR 91i:68138 Zl