Computing skew left braces of small orders

TL;DR

This paper enhances an existing algorithm to compute all non-isomorphic skew left braces, enumerates structures up to order 868, and proposes new conjectures based on the data.

Contribution

It improves an algorithm for computing skew left braces and provides extensive enumeration data up to order 868, facilitating future research.

Findings

Enumerated skew left braces up to order 868

Proposed new conjectures based on enumeration data

Enhanced algorithm for computing non-isomorphic skew left braces

Abstract

We improve Algorithm 5.1 of [Math. Comp. {\bf 86} (2017), 2519-2534] for computing all non-isomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we state some conjectures for further research.

Tables

Table 1. Table 1. Some missing values from [ 22 ]

$n$	$32$	$54$	$64$	$72$	$80$	$81$	$96$	$108$
$b\left(n\right)$	$25281$	$80$	$?$	$489$	$1985$	$804$	$195971$	$494$
$s\left(n\right)$	$1223061$	$1028$	$?$	$17790$	$74120$	$8436$	$?$	$11223$
$n$	$112$	$120$	$126$	$128$	$136$	$144$	$147$	$150$
$b\left(n\right)$	$1671$	$395$	$36$	$?$	$108$	$10215$	$9$	$19$
$s\left(n\right)$	$65485$	$22711$	$990$	$?$	$986$	$3013486$	$123$	$401$
$n$	$152$	$158$	$160$	$162$	$164$	$165$	$166$	$168$
$b\left(n\right)$	$90$	$2$	$209513$	$1374$	$11$	$2$	$2$	$443$
$s\left(n\right)$	$800$	$6$	$?$	$45472$	$43$	$12$	$6$	$28505$

Table 2. Table 2. Further Computations

$n$	$169$	$170$	$171$	$172$	$173$	$174$	$175$	$176$	$177$	$178$
$b\left(n\right)$	$4$	$4$	$14$	$9$	$1$	$4$	$4$	$1670$	$1$	$2$
$s\left(n\right)$	$4$	$36$	$80$	$29$	$1$	$36$	$4$	$65466$	$1$	$6$
$n$	$179$	$180$	$181$	$182$	$183$	$184$	$185$	$186$	$187$	$188$
$b\left(n\right)$	$1$	$129$	$1$	$4$	$2$	$90$	$1$	$6$	$1$	$9$
$s\left(n\right)$	$1$	$5849$	$1$	$36$	$8$	$800$	$1$	$78$	$1$	$29$
$n$	$189$	$190$	$191$	$192$	$193$	$194$	$195$	$196$	$197$	$198$
$b\left(n\right)$	$165$	$4$	$1$	$?$	$1$	$2$	$2$	$41$	$1$	$16$
$s\left(n\right)$	$4560$	$36$	$1$	$?$	$1$	$6$	$8$	$389$	$1$	$294$
$n$	$199$	$200$	$201$	$202$	$203$	$204$	$205$	$206$	$207$	$208$
$b\left(n\right)$	$1$	$568$	$2$	$2$	$2$	$28$	$2$	$2$	$4$	$1984$
$s\left(n\right)$	$1$	$23471$	$8$	$6$	$16$	$410$	$12$	$6$	$4$	$74104$
$n$	$209$	$210$	$211$	$212$	$213$	$214$	$215$	$216$	$217$	$218$
$b\left(n\right)$	$1$	$12$	$1$	$11$	$1$	$2$	$1$	$5308$	$1$	$2$
$s\left(n\right)$	$1$	$468$	$1$	$43$	$1$	$6$	$1$	$523768$	$1$	$6$
$n$	$219$	$220$	$221$	$222$	$223$	$224$	$225$	$226$	227	$228$
$b\left(n\right)$	$2$	$36$	$1$	$6$	$1$	$195483$	$21$	$2$	$1$	$34$
$s\left(n\right)$	$8$	$702$	$1$	$78$	$1$	$?$	$61$	$6$	$1$	$606$
$n$	$229$	$230$	$231$	$232$	$233$	$234$	$235$	$256$	$237$	$238$
$b\left(n\right)$	$1$	$4$	$2$	$106$	$1$	$36$	$1$	$9$	$2$	$4$
$s\left(n\right)$	$1$	$36$	$8$	$944$	$1$	$990$	$1$	$29$	$8$	$36$
$n$	$239$	$240$	$241$	$242$	$243$	$244$	$245$	$246$	$247$	$248$
$b\left(n\right)$	$1$	$10518$	$1$	$8$	$598065$	$11$	$4$	$4$	$1$	$90$
$s\left(n\right)$	$1$	$4642485$	$1$	$57$	$>27447027$	$43$	$4$	$36$	$1$	$800$
$n$	$249$	$250$	$251$	$252$	$253$	$254$	$255$	$256$	$257$	$258$
$b\left(n\right)$	$1$	$104$	$1$	$229$	$2$	$2$	$1$	$?$	$1$	$6$
$s\left(n\right)$	$1$	$1492$	$1$	$11541$	$24$	$6$	$1$	$?$	$1$	$78$
$n$	$259$	$260$	$261$	$262$	$263$	$264$	$265$	$266$	$267$	$268$
$b\left(n\right)$	$1$	$35$	$4$	$2$	$1$	$345$	$1$	$4$	$1$	$9$
$s\left(n\right)$	$1$	$739$	$4$	$6$	$1$	$20231$	$1$	$36$	$1$	$29$
$n$	$269$	$270$	$271$	$272$	$273$	$274$	$275$	$276$	$277$	$278$
$b\left(n\right)$	$1$	$160$	$1$	$2014$	$5$	$2$	$13$	$24$	$1$	$2$
$s\left(n\right)$	$1$	$6168$	$1$	$74960$	$113$	$6$	$93$	$324$	$1$	$6$
$n$	$279$	$280$	$281$	$282$	$283$	$284$	$285$	$286$	$287$	$288$
$b\left(n\right)$	$11$	$385$	$1$	$4$	$1$	$9$	$2$	$4$	$1$	$1392959$
$s\left(n\right)$	$47$	$22295$	$1$	$36$	$1$	$29$	$8$	$36$	$1$	$?$
$n$	$289$	$290$	$291$	$292$	$293$	$294$	$295$	$296$	$297$	$298$
$b\left(n\right)$	$4$	$4$	$2$	$11$	$1$	$31$	$1$	$106$	$37$	$2$
$s\left(n\right)$	$4$	$36$	$8$	$43$	$1$	$2152$	$1$	$944$	$101$	$6$
$n$	$299$	$300$	$301$	$302$	$303$	$304$	$305$	$306$	$307$	$308$
$b\left(n\right)$	$1$	$152$	$2$	$2$	$1$	$1670$	$2$	16	$1$	$23$
$s\left(n\right)$	$1$	$8222$	$16$	$6$	$1$	$65466$	$12$	$294$	$1$	$311$
$n$	$309$	$310$	$311$	$312$	$313$	$314$	$315$	$316$	$317$	$318$
$b\left(n\right)$	$2$	$6$	$1$	$507$	$1$	$2$	$11$	$9$	$1$	$4$
$s\left(n\right)$	$8$	$94$	$1$	$32075$	$1$	$6$	$47$	$29$	$1$	$36$
$n$	$319$	$320$	$321$	$322$	$323$	$324$	$325$	$326$	$327$	$328$
$b\left(n\right)$	$1$	$?$	$1$	$4$	$1$	$10225$	$4$	$2$	$2$	$108$
$s\left(n\right)$	$1$	$?$	$1$	$36$	$1$	$?$	$4$	$6$	$8$	$986$
$n$	$329$	$330$	$331$	$332$	$333$	$334$	$335$	$336$	$337$	$338$
$b\left(n\right)$	$1$	$12$	$1$	$9$	$14$	$2$	$1$	$10990$	$1$	$8$
$s\left(n\right)$	$1$	$564$	$1$	$29$	$80$	$6$	$1$	$5247711$	$1$	$59$
$n$	$339$	$340$	$341$	$342$	$343$	$344$	$345$	$346$	$347$	$348$
$b\left(n\right)$	$1$	$35$	$1$	$42$	$61$	$90$	$1$	$2$	$1$	$28$
$s\left(n\right)$	$1$	$739$	$1$	$1164$	$373$	$800$	$1$	$6$	$1$	$410$

Table 3. Table 3. Further Computations

$n$	$349$	$350$	$351$	$352$	$353$	$354$	$355$	$356$	$357$	$358$
$b\left(n\right)$	$1$	$16$	$166$	$195479$	$1$	$4$	$2$	$11$	$2$	$2$
$s\left(n\right)$	$1$	$306$	$4591$	$?$	$1$	$36$	$12$	$43$	$8$	$6$
$n$	$359$	$360$	$361$	$362$	$363$	$364$	$365$	$366$	$367$	$368$
$b\left(n\right)$	$1$	$2035$	$4$	$2$	$5$	$27$	$1$	$6$	$1$	$1670$
$s\left(n\right)$	$1$	$535713$	$4$	$6$	$20$	$395$	$1$	$78$	$1$	$65466$
$n$	$369$	$370$	$371$	$372$	$373$	$374$	$375$	$376$	$377$	$378$
$b\left(n\right)$	$4$	$4$	$1$	$34$	$1$	$4$	$54$	$90$	$1$	$548$
$s\left(n\right)$	$4$	$36$	$1$	$606$	$1$	$36$	$253$	$800$	$1$	$47244$
$n$	$379$	$380$	$381$	$382$	$383$	$384$	$385$	$386$	$387$	$388$
$b\left(n\right)$	$1$	$27$	$2$	$2$	$1$	$?$	$2$	$2$	$11$	$11$
$s\left(n\right)$	$1$	$395$	$8$	$6$	$1$	$?$	$12$	$6$	$47$	$43$
$n$	$389$	$390$	$391$	$392$	$393$	$394$	$395$	$396$	$397$	$398$
$b\left(n\right)$	$1$	$12$	$1$	$463$	$1$	$2$	$1$	$111$	$1$	$2$
$s\left(n\right)$	$1$	$468$	$1$	$18078$	$1$	$6$	$1$	$4985$	$1$	$6$
$n$	$399$	$400$	$401$	$402$	$403$	$404$	$405$	$406$	$407$	$408$
$b\left(n\right)$	$5$	$12744$	$1$	$6$	$1$	$11$	$805$	$6$	$1$	$399$
$s\left(n\right)$	$113$	$3618636$	$1$	$78$	$1$	$43$	$8453$	$110$	$1$	$22923$
$n$	$409$	$410$	$411$	$412$	$413$	$414$	$415$	$416$	$417$	$418$
$b\left(n\right)$	$1$	$6$	$1$	$9$	$1$	$16$	$1$	$209507$	$2$	$4$
$s\left(n\right)$	$1$	$94$	$1$	$29$	$1$	$294$	$1$	$?$	$8$	$36$
$n$	$419$	$420$	$421$	$422$	$423$	$424$	$425$	$426$	$427$	$428$
$b\left(n\right)$	$1$	$104$	$1$	$2$	$4$	$106$	$4$	$4$	$1$	$9$
$s\left(n\right)$	$1$	$9052$	$1$	$6$	$4$	$944$	$4$	$36$	$1$	$29$
$n$	$429$	$430$	$431$	$432$	$433$	$434$	$435$	$436$	$437$	$438$
$b\left(n\right)$	$2$	$4$	$1$	$115708$	$1$	$4$	$1$	$11$	$1$	$6$
$s\left(n\right)$	$8$	$36$	$1$	$?$	$1$	$36$	$1$	$43$	$1$	$78$
$n$	$439$	$440$	$441$	$442$	$443$	$444$	$445$	$446$	$447$	$448$
$b\left(n\right)$	$1$	$474$	$55$	$4$	$1$	$40$	$1$	$2$	$1$	$?$
$s\left(n\right)$	$1$	$31970$	$1110$	$36$	$1$	$782$	$1$	$6$	$1$	$?$
$n$	$449$	$450$	$451$	$452$	$453$	$454$	$455$	$456$	$457$	$458$
$b\left(n\right)$	$1$	$82$	$1$	$11$	$2$	$2$	$1$	$441$	$1$	$2$
$s\left(n\right)$	$1$	$3797$	$1$	$43$	$8$	$6$	$1$	$28447$	$1$	$6$
$n$	$459$	$460$	$461$	$462$	$463$	$464$	$465$	$466$	$467$	$468$
$b\left(n\right)$	$37$	$27$	$1$	$12$	$1$	$1984$	$4$	$2$	$1$	$267$
$s\left(n\right)$	$101$	$395$	$1$	$468$	$1$	$74104$	$66$	$6$	$1$	$13941$
$n$	$469$	$470$	$471$	$472$	$473$	$474$	$475$	$476$	$477$	$478$
$b\left(n\right)$	$1$	$4$	$2$	$90$	$1$	$6$	$4$	$27$	$4$	$2$
$s\left(n\right)$	$1$	$36$	$8$	$800$	$1$	$78$	$4$	$395$	$4$	$6$
$n$	$479$	$480$	$481$	$482$	$483$	$484$	$485$	$486$	$487$	$488$
$b\left(n\right)$	$1$	$?$	$1$	$2$	$2$	$41$	$1$	$639775$	$1$	$106$
$s\left(n\right)$	$1$	$?$	$1$	$6$	$8$	$421$	$1$	$?$	$1$	$944$
$n$	$489$	$490$	$491$	$492$	$493$	$494$	$495$	$496$	$497$	$498$
$b\left(n\right)$	$2$	$16$	$1$	$28$	$1$	$4$	$8$	$1670$	$2$	$4$
$s\left(n\right)$	$8$	$318$	$1$	$410$	$1$	$36$	$48$	$65466$	$16$	$36$
$n$	$499$	$500$	$501$	$502$	$503$	$504$	$505$	$506$	$507$	$508$
$b\left(n\right)$	$1$	$634$	$1$	$2$	$1$	$3249$	$2$	$6$	$9$	$9$
$s\left(n\right)$	$1$	$21252$	$1$	$6$	$1$	$871013$	$12$	$142$	$135$	$29$
$n$	$509$	$510$	$511$	$512$	$513$	$514$	$515$	$516$	$517$	$518$
$b\left(n\right)$	$1$	$8$	$1$	$?$	$189$	$2$	$1$	$34$	$1$	$4$
$s\left(n\right)$	$1$	$216$	$1$	$?$	$5055$	$6$	$1$	$606$	$1$	$36$
$n$	$519$	$520$	$521$	$522$	$523$	$524$	$525$	$526$	$527$	$528$
$b\left(n\right)$	$1$	$484$	$1$	$16$	$1$	$9$	$10$	$2$	$1$	$9274$
$s\left(n\right)$	$1$	$28714$	$1$	$294$	$1$	$29$	$112$	$6$	$1$	$4381956$

Table 4. Table 4. Further Computations

$n$	$529$	$530$	$531$	$532$	$533$	$534$	$535$	$536$	$537$	$538$
$b\left(n\right)$	$4$	$4$	$4$	$23$	$1$	$4$	$1$	$90$	$1$	$2$
$s\left(n\right)$	$4$	$36$	$4$	$311$	$1$	$36$	$1$	$800$	$1$	$6$
$n$	$539$	$540$	$541$	$542$	$543$	$544$	$545$	$546$	$547$	$548$
$b\left(n\right)$	$4$	$1342$	$1$	$2$	$2$	$210043$	$1$	$24$	$1$	$11$
$s\left(n\right)$	$4$	$148151$	$1$	$6$	$8$	$?$	$1$	$2664$	$1$	$43$
$n$	$549$	$550$	$551$	$552$	$553$	$554$	$555$	$556$	$557$	$558$
$b\left(n\right)$	$11$	$40$	$1$	$345$	$1$	$2$	$2$	$9$	$1$	$36$
$s\left(n\right)$	$47$	$1370$	$1$	$20231$	$1$	$6$	$8$	$29$	$1$	$990$
$n$	$559$	$560$	$561$	$562$	$563$	$564$	$565$	$566$	$567$	$568$
$b\left(n\right)$	$1$	$10423$	$1$	$2$	$1$	$24$	$1$	$2$	$7196$	$90$
$s\left(n\right)$	$1$	$4633376$	$1$	$6$	$1$	$324$	$1$	$6$	$2253564$	$800$
$n$	$569$	$570$	$571$	$572$	$573$	$574$	$575$	$576$	$577$	$578$
$b\left(n\right)$	$1$	$12$	$1$	$27$	$1$	$4$	$4$	$?$	$1$	$8$
$s\left(n\right)$	$1$	$468$	$1$	$395$	$1$	$36$	$4$	$?$	$1$	$63$
$n$	$579$	$580$	$581$	$582$	$583$	$584$	$585$	$586$	$587$	$588$
$b\left(n\right)$	$2$	$35$	$1$	$6$	$1$	$108$	$11$	$2$	$1$	$202$
$s\left(n\right)$	$8$	$739$	$1$	$78$	$1$	$986$	$47$	$6$	$1$	$21836$
$n$	$589$	$590$	$591$	$592$	$593$	$594$	$595$	$596$	$597$	$598$
$b\left(n\right)$	$1$	$4$	$1$	$1984$	$1$	$160$	$1$	$11$	$2$	$4$
$s\left(n\right)$	$1$	$36$	$1$	$74104$	$1$	$6168$	$1$	$43$	$8$	$36$
$n$	$599$	$600$	$601$	$602$	$603$	$604$	$605$	$606$	$607$	$608$
$b\left(n\right)$	$1$	$2413$	$1$	$6$	$11$	$9$	$10$	$4$	$1$	$195479$
$s\left(n\right)$	$1$	$659897$	$1$	$110$	$47$	$29$	$409$	$36$	$1$	$?$
$n$	$609$	$610$	$611$	$612$	$613$	$614$	$615$	$616$	$617$	$618$
$b\left(n\right)$	$3$	$6$	$1$	$129$	$1$	$2$	$2$	$335$	$1$	$6$
$s\left(n\right)$	$25$	$94$	$1$	$5835$	$1$	$6$	$12$	$19885$	$1$	$78$
$n$	$619$	$620$	$621$	$622$	$623$	$624$	$625$	$626$	$627$	$628$
$b\left(n\right)$	$1$	$36$	$37$	$2$	$1$	$12547$	$2308$	$2$	$2$	$11$
$s\left(n\right)$	$1$	$702$	$101$	$6$	$1$	$5595183$	$69032$	$6$	$8$	$43$
$n$	$629$	$630$	$631$	$632$	$633$	$634$	$635$	$636$	$637$	$638$
$b\left(n\right)$	$1$	$72$	$1$	$90$	$2$	$2$	$1$	$28$	$4$	$4$
$s\left(n\right)$	$1$	$5940$	$1$	$800$	$8$	$6$	$1$	$410$	$4$	$36$
$n$	$639$	$640$	$641$	$642$	$643$	$644$	$645$	$646$	$647$	$648$
$b\left(n\right)$	$4$	$?$	$1$	$4$	$1$	$23$	$2$	$4$	$1$	$91071$
$s\left(n\right)$	$4$	$?$	$1$	$36$	$1$	$311$	$8$	$36$	$1$	$?$
$n$	$649$	$650$	$651$	$652$	$653$	$654$	$655$	$656$	$657$	$658$
$b\left(n\right)$	$1$	$16$	$5$	$9$	$1$	$6$	$2$	$2010$	$14$	$4$
$s\left(n\right)$	$1$	$306$	$113$	$29$	$1$	$78$	$12$	$74860$	$80$	$36$
$n$	$659$	$660$	$661$	$662$	$663$	$664$	$665$	$666$	$667$	$668$
$b\left(n\right)$	$1$	$100$	$1$	$2$	$2$	$90$	$1$	$42$	$1$	$9$
$s\left(n\right)$	$1$	$9346$	$1$	$6$	$8$	$800$	$1$	$1164$	$1$	$29$
$n$	$669$	$670$	$671$	$672$	$673$	$674$	$675$	$676$	$677$	$678$
$b\left(n\right)$	$2$	$4$	$1$	$?$	$1$	$2$	$232$	$51$	$1$	$4$
$s\left(n\right)$	$8$	$36$	$1$	$?$	$1$	$6$	$3682$	$791$	$1$	$36$
$n$	$679$	$680$	$681$	$682$	$683$	$684$	$685$	$686$	$687$	$688$
$b\left(n\right)$	$1$	$492$	$1$	$4$	$1$	$259$	$1$	$128$	$2$	$1670$
$s\left(n\right)$	$1$	$29698$	$1$	$36$	$1$	$12723$	$1$	$2084$	$8$	$65466$
$n$	$689$	$690$	$691$	$692$	$693$	$694$	$695$	$696$	$697$	$698$
$b\left(n\right)$	$2$	$8$	$1$	$11$	$11$	$2$	$1$	$395$	$1$	$2$
$s\left(n\right)$	$28$	$216$	$1$	$43$	$47$	$6$	$1$	$22667$	$1$	$6$
$n$	$699$	$700$	$701$	$702$	$7703$	$704$	$705$	$706$	$707$	$708$
$b\left(n\right)$	$1$	$126$	$1$	$550$	$1$	$?$	$1$	$2$	$1$	$24$
$s\left(n\right)$	$1$	$7102$	$1$	$47374$	$1$	$?$	$1$	$6$	$1$	$324$

Table 5. Table 5. Further Computations

$n$	$709$	$710$	$711$	$712$	$713$	$714$	$715$	$716$	$717$	$718$
$b\left(n\right)$	$1$	$6$	$11$	$108$	$1$	$12$	$2$	$9$	$1$	$2$
$s\left(n\right)$	$1$	$94$	$47$	$986$	$1$	$468$	$12$	$29$	$1$	$6$
$n$	$719$	$720$	$721$	$722$	$723$	$724$	$725$	$726$	$727$	$728$
$b\left(n\right)$	$1$	$65074$	$1$	$8$	$2$	$11$	$4$	$19$	$1$	$385$
$s\left(n\right)$	$1$	$?$	$1$	$65$	$8$	$43$	$4$	$466$	$1$	$22295$
$n$	$729$	$730$	$731$	$732$	$733$	$734$	$735$	$736$	$737$	$738$
$b\left(n\right)$	$?$	$4$	$1$	$40$	$1$	$2$	$9$	$195479$	$2$	$16$
$s\left(n\right)$	$?$	$36$	$1$	$782$	$1$	$6$	$123$	$?$	$24$	$294$
$n$	$739$	$740$	$741$	$742$	$743$	$744$	$745$	$746$	$747$	$748$
$b\left(n\right)$	$1$	$35$	$5$	$4$	$1$	$441$	$1$	$2$	$4$	$27$
$s\left(n\right)$	$1$	$739$	$113$	$36$	$1$	$28447$	$1$	$6$	$4$	$395$
$n$	$749$	$750$	$751$	$752$	$753$	$754$	$755$	$756$	$757$	$758$
$b\left(n\right)$	$1$	$224$	$1$	$1670$	$1$	$4$	$2$	$3757$	$1$	$2$
$s\left(n\right)$	$1$	$10001$	$1$	$65466$	$1$	$36$	$12$	$794193$	$1$	$6$
$n$	$759$	$760$	$761$	$762$	$763$	$764$	$765$	$766$	$767$	$768$
$b\left(n\right)$	$2$	$384$	$1$	$6$	$1$	$9$	$4$	$2$	$1$	$?$
$s\left(n\right)$	$24$	$22278$	$1$	$78$	$1$	$29$	$4$	$6$	$1$	$?$
$n$	$769$	$770$	$771$	$772$	$773$	$774$	$775$	$776$	$777$	$778$
$b\left(n\right)$	$1$	$12$	$1$	$11$	$1$	$36$	$13$	$108$	$5$	$2$
$s\left(n\right)$	$1$	$564$	$1$	$43$	$1$	$990$	$93$	$986$	$113$	$6$
$n$	$779$	$780$	$781$	$782$	$783$	$784$	$785$	$786$	$787$	$788$
$b\left(n\right)$	$1$	$128$	$1$	$4$	$37$	$9998$	$1$	$4$	$1$	$11$
$s\left(n\right)$	$1$	$13320$	$1$	$36$	$101$	$3074483$	$1$	$36$	$1$	$43$
$n$	$789$	$790$	$791$	$792$	$793$	$794$	$795$	$796$	$797$	$798$
$b\left(n\right)$	$1$	$4$	$2$	$1771$	$1$	$2$	$1$	$9$	$1$	$24$
$s\left(n\right)$	$1$	$36$	$16$	$484183$	$1$	$6$	$1$	$29$	$1$	$2664$
$n$	$799$	$800$	$801$	$802$	$803$	$804$	$805$	$806$	$807$	$808$
$b\left(n\right)$	$1$	$?$	$4$	$2$	$1$	$34$	$1$	$4$	$1$	$106$
$s\left(n\right)$	$1$	$?$	$4$	$6$	$1$	$606$	$1$	$36$	$1$	$944$
$n$	$809$	$810$	$811$	$812$	$813$	$814$	$815$	$816$	$817$	$818$
$b\left(n\right)$	$1$	$2751$	$1$	$38$	$2$	$4$	$1$	$10604$	$1$	$2$
$s\left(n\right)$	$1$	$272960$	$1$	$920$	$8$	$36$	$1$	$4658179$	$1$	$6$
$n$	$819$	$820$	$821$	$822$	$823$	$824$	$825$	$826$	$827$	$828$
$b\left(n\right)$	$41$	$46$	$1$	$4$	$1$	$90$	$14$	$4$	$1$	$111$
$s\left(n\right)$	$1337$	$1212$	$1$	$36$	$1$	$800$	$105$	$36$	$1$	$4985$
$n$	$829$	$830$	$831$	$832$	$833$	$834$	$835$	$836$	$837$	$838$
$b\left(n\right)$	$1$	$4$	$2$	$?$	$4$	$6$	$1$	$23$	$165$	$2$
$s\left(n\right)$	$1$	$36$	$8$	$?$	$4$	$78$	$1$	$311$	$4560$	$6$
$n$	$839$	$840$	$841$	$842$	$843$	$844$	$845$	$846$	$847$	$848$
$b\left(n\right)$	$1$	$1933$	$4$	$2$	$1$	$9$	$4$	$16$	$4$	$1984$
$s\left(n\right)$	$1$	$878779$	$4$	$6$	$1$	$27$	$4$	$294$	$4$	$74104$
$n$	$849$	$850$	$851$	$852$	$853$	$854$	$855$	$856$	$857$	$858$
$b\left(n\right)$	$2$	$16$	$1$	$24$	$1$	$4$	$14$	$90$	$1$	$12$
$s\left(n\right)$	$8$	$306$	$1$	$324$	$1$	$36$	$80$	$800$	$1$	$468$
$n$	$859$	$860$	$861$	$862$	$863$	$864$	$865$	$866$	$867$	$868$
$b\left(n\right)$	$1$	$27$	$2$	$2$	$1$	$?$	$1$	$2$	$5$	$23$
$s\left(n\right)$	$1$	$395$	$8$	$6$	$1$	$?$	$1$	$6$	$26$	$311$

Table 6. Table 6. Enumerations of left braces of order 64

$GroupId$	$[64,1]$	$[64,2]$	$[64,26]$	$[64,50]$	$[64,55]$	$[64,83]$
$Number$	$10$	$11354$	$2742$	$142$	$?$	$734410$
$GroupId$	$[64,183]$	$[64,192]$	$[64,246]$	$[64,260]$	$[64,267]$
$Number$	$3124$	$?$	$253350$	$2189661$	$58558$

Table 7. Table 7. Enumerations of left braces of order 480

$GroupId$	$[480,4]$	$[480,199]$	$[480,212]$	$[480,919]$	$[480,934]$	$[480,1180]$	$[480,1213]$
$Number$	$128$	$?$	$4928$	$958965$	$99970$	$?$	$39650$

Table 8. Table 8. Time comparison on skew left braces of order 32

$GroupIdoftheadditivegroup$	$[32,23]$	$[32,24]$	$[32,25]$	$[32,28]$	$[32,29]$	$[32,30]$
$Numberofskewbracestructures$	$39488$	$70400$	$138336$	$138336$	$138336$	$137526$
$TimeonAlgorithm\mathrm{\hspace{0.17em}5.1}\left[13\right]$	$11238$	$9808$	$18720$	$10193$	$10083$	$34005$
$TimeonAlgorithm\text{<a><span>2.3</span></a>}$	$539$	$709$	$1905$	$4308$	$3135$	$4658$
$GroupIdoftheadditivegroup$	$[32,31]$	$[32,32]$	$[32,33]$	$[32,45]$	$[32,47]$	$[32,51]$
$Numberofskewbracestructures$	$70944$	$69236$	$91008$	$8015$	$7870$	$744$
$TimeonAlgorithm\mathrm{\hspace{0.17em}5.1}\left[13\right]$	$14568$	$18342$	$17222$	$130942$	$28848$	$\#$
$TimeonAlgorithm\text{<a><span>2.3</span></a>}$	$4797$	$9302$	$8869$	$30$	$68$	$8$

Equations

$$g_1\circ (g_2+g_3)=(g_1\circ g_2)-g_1+(g_1\circ g_3)$$

$$R_{12}{R}_{23}{R}_{12}=R_{23}{R}_{12}{R}_{23}$$

$$R(x,y)=\big{(}\sigma_{x}(y),\tau_{y}(x)\big{)},~{}x,y\in X$$

$$\mathrm{Hol}(G):=\mathrm{Aut}(G)⋉G,$$

$$(\alpha ,g)(\beta ,h)=(\alpha \beta ,g\alpha (h)).$$

$$g^{(\alpha,h)}=\pi_{2}\big{(}(\alpha,h)(1,g)\big{)}=h\alpha(g)$$

$$(\alpha_{1},h_{1})(\alpha,h)=\big{(}\alpha_{1}\alpha,h_{1}\alpha_{1}(h)\big{)}=(\alpha_{1}\alpha,1).$$

$$\mathcal{G}^{(\beta,1)}=\mathcal{G}^{(\alpha_{1},h_{1})(\alpha,h)}=\big{(}\mathcal{G}^{(\alpha_{1},h_{1})}\big{)}^{(\alpha,h)}=\mathcal{G}^{(\alpha,h)}.$$

$$b(4q)=\{\begin{array}{ll}9,& \text{if}q\equiv 3\mathrm{mod}4\\ 11,& \text{if }q\equiv 1\mathrm{mod}4\end{array}$$

$$b(p^{2}q)=\{\begin{array}{ll}4,& \text{if}3\nmid p-1\\ p+8,& \text{if}3\mid p-1\text{and}9\nmid p-1\\ 2p+8,& \text{if }9\mid p-1.\end{array}$$

$$s(4q)=\{\begin{array}{ll}29,& \text{if}q\equiv 3\mathrm{mod}4\\ 43,& \text{if }q\equiv 1\mathrm{mod}4\end{array}$$

$$s(p^{2}q)=\{\begin{array}{ll}4,& \text{if}p\nmid q-1\\ 2p^2+7p+8,& \text{if}p\mid q-1\text{and}{p}^2\nmid q-1\\ 6p^2+6p+8,& \text{if }{p}^2\mid q-1.\end{array}$$

$$b(8p)=\{\begin{array}{ll}90,& \text{if}p\equiv 3,7\mathrm{mod}8\\ 106,& \text{if}p\equiv 5\mathrm{mod}8\\ 108,& \text{if }p\equiv 1\mathrm{mod}8.\end{array}$$

$$s(8p)=\{\begin{array}{ll}800,& \text{if}p\equiv 3,7\mathrm{mod}8\\ 944,& \text{if}p\equiv 5\mathrm{mod}8\\ 986,& \text{if }p\equiv 1\mathrm{mod}8.\end{array}$$

$$b(12p)=\{\begin{array}{ll}24,& \text{if}p\equiv 11\mathrm{mod}12\\ 28,& \text{if}p\equiv 5\mathrm{mod}12\\ 34,& \text{if}p\equiv 7\mathrm{mod}12\\ 40,& \text{if }p\equiv 1\mathrm{mod}12.\end{array}$$

$$s(12p)=\{\begin{array}{ll}324,& \text{if}p\equiv 11\mathrm{mod}12\\ 410,& \text{if}p\equiv 5\mathrm{mod}12\\ 606,& \text{if}p\equiv 7\mathrm{mod}12\\ 782,& \text{if }p\equiv 1\mathrm{mod}12.\end{array}$$

$$b(2pq)=\{\begin{array}{ll}4,& \text{if}p\nmid q-1\\ 6,& \text{if }p\mid q-1\end{array}$$

$$s(2pq)=\{\begin{array}{ll}36,& \text{if}p\nmid q-1\\ 8p+54,& \text{if}p\mid q-1.\end{array}$$

Full text

1. Introduction

A triple $(G,+,\circ)$, where $(G,+)$ and $(G,\circ)$ are (not necessarily abelian) groups, is said to be a skew left brace if

[TABLE]

for all $g_{1},g_{2},g_{3}\in G$, where $-g_{1}$ denotes the inverse of $g_{1}$ in $(G,+)$. We call $(G,+)$ the additive group and $(G,\circ)$ the multiplicative group of the skew left brace $(G,+,\circ)$. A skew left brace $(G,+,\circ)$ is said to be a left brace if $(G,+)$ is an abelian group. The concept of left braces was introduced by Rump [18] in 2007 in connection with non-degenerate involutive set theoretic solutions of the quantum Yang-Baxter equations. Thereafter the subject received a tremendous attention of the mathematical community; see [3, 5, 19, 20] and the references therein. Interest in the study of set theoretic solutions of the quantum Yang-Baxter equations was intrigued by the paper [11] of Drinfeld, published in 1992.

Let $X$ be an arbitrary set and $R:X\times X\to X\times X$ a bijective map. Recall that the pair $(X,R)$ is said to be a set theoretic solution of the Yang-Baxter equation if

[TABLE]

holds in the set of all maps from $X\times X\times X$ to itself, where $R_{ij}$ is just $R$ acting on the $i$th and $j$th components of $X\times X\times X$ and identity on the remaining one. Let us write

[TABLE]

with $\sigma_{x}$ and $\tau_{y}$ component maps from $X$ to itself.

A solution $(X,R)$ is said to be non-degenerate if the component maps $\sigma_{x}$ and $\tau_{y}$ are bijections on $X$ for all $x,y\in X$. It is said to be involutive if $R^{2}$ is the identity map. The study of non-degenerate set theoretic solutions of the quantum Yang-Baxter equations has been extensively taken up, e. g., [6, 9, 13, 16, 21] to mention a few.

The concept of skew left brace was introduced by Guarnieri and Vendramin [15] in 2017 in connection with non-involutive non-degenerate set theoretic solutions of the quantum Yang-Baxter equations. They invented an algorithm, by generalising a result of Bachiller [2] for computing all skew left braces of a given order. They themselves computed left braces and skew left braces of lot of groups upto order 120. Vendramin [22] extended the number upto 168 with some exceptions. All these computations are done using computer algebra systems MAGMA [4] and GAP [14] using the algorithm invented in [15]. For more work on skew braces see [7, 8, 17].

This article aims at filling up the gaps in the table produced in [22] to some extent and making further computations for larger orders. An ingenious observation on regular subgroups of the holomorph of a given finite group allows us to improve the algorithm obtained in [15], which substantially enhances the performance of MAGMA computation. The improved algorithm, actually, avoids an expensive calculation in the existing algorithm. We compute the number of non-isomorphic left braces and skew left braces of orders upto 868 except certain cases (mainly when the order is a multiple of 32). These results settle [22, Problem 13] and [15, Problem 6.1]. The computations will help in building a database of left braces and skew left braces, which in turn will greatly enrich the library of solutions of the quantum Yang-Baxter equation. On the basis of our computation, we suggest some conjectures for further research.

It is striking that there are more than a million skew brace structures of order $2^{5}$ and more than 20 millions skew brace structures of order $3^{5}$. The reader will encounter many more surprises while going through the tables. We have used MAGMA on a computer with 3.5 GHz 6-Core Intel Xeon E5 processor and 64 GB memory for these computations.

2. Regular subgroups

Let $G$ be a group, which acts on a set $X$. The action of an element $g\in G$ on an element $x\in X$ is denoted by $x^{g}$. A subgroup $H$ of $G$ is said to be action-closed if for each pair $(g,x)\in G\times X$, there exists an element $h\in H$ such that $x^{g}=x^{h}$. By $H$* - conjugacy class* of $x\in G$, we mean $\{x^{h}\mid h\in H\}$. For $g,h\in G$, we write the conjugate of $g$ by $h$ as $g^{h}=h^{-1}gh$.

Let $G$ be a group and $\operatorname{Symm}(G)$ be the symmetric group on the set $G$. Recall that a subgroup $\mathcal{G}$ of $\operatorname{Symm}(G)$ is said to be regular if $\mathcal{G}$-action on $G$ is free and transitive. By a free action we here mean that for any element $g\in G$, its stabilizer in $\mathcal{G}$ is the trivial subgroup. Observe that when $G$ is finite, any regular subgroup of $\operatorname{Symm}(G)$ is of order $|G|$.

For a group $G$, $\operatorname{Hol}(G)$ denotes the holomorph of $G$, which is defined as the semidirect product of $G$ with $\operatorname{Aut}(G)$, the automorphism group of $G$. So

[TABLE]

where the product in $\operatorname{Hol}(G)$ is given by

[TABLE]

Notice that $\operatorname{Hol}(G)$ acts on $G$ transitively under the following action:

[TABLE]

for all $\alpha\in\operatorname{Aut}(G)$ and $g,h\in G$, where $\pi_{2}:\operatorname{Hol}(G)\to G$ is the projection map given by \pi_{2}\big{(}(\alpha,g)\big{)}=g. It follows that the stabilizer of any element of $G$ in $\operatorname{Hol}(G)$ is isomorphic to $\operatorname{Aut}(G)$.

Let $\mathcal{G}$ be a regular subgroup of $\operatorname{Hol}(G)$. Then it is not difficult to see that for each $g\in G$, there exists a unique element $(\alpha,h)\in\mathcal{G}$ such that $g^{(\alpha,h)}=h\alpha(g)=1$. Let $\operatorname{Reg}(G)$ denote the set of all regular subgroups of $\operatorname{Hol}(G)$. Then $\operatorname{Hol}(G)$ acts on $\operatorname{Reg}(G)$ by conjugation. With this setting, we have the following easy observation, which plays a key role in what follows.

Lemma 2.1**.**

$\operatorname{Aut}(G)$, as a subgroup of $\operatorname{Hol}(G)$, is action-closed with respect to the conjugation action of $\operatorname{Hol}(G)$ on $\operatorname{Reg}(G)$.

Proof.

Let $\mathcal{G}\in\operatorname{Reg}(G)$ and $(\alpha,h)\in\operatorname{Hol}(G)$. Then there exists an element $(\alpha_{1},h_{1})\in\mathcal{G}$ such that $h^{(\alpha_{1},h_{1})}=h_{1}\alpha_{1}(h)=1$. Notice that

[TABLE]

Let $\beta:=\alpha_{1}\alpha$, which lies in $\operatorname{Aut}(G)$. Thus,

[TABLE]

Proof is now complete. ∎

The preceding lemma enables us to get the following generalization of [15, Proposition 4.3].

Theorem 2.2**.**

Let $(G,+)$ be a group. Then non-isomorphic skew left braces $(G,+,\circ)$ are in bijective correspondence with conjugacy classes of regular subgroups in $\operatorname{Hol}(G,+)$. Moreover, if $G$ is a $p$-group for some prime $p$, then non-isomorphic skew left brace structures over $G$ are in bijective correspondence with $\operatorname{Aut}(G)$ - conjugacy classes of regular subgroups of any Sylow $p$-subgroup of $\operatorname{Hol}(G,+)$.

Proof.

The first assertion follows from [15, Proposition 4.3] along with Lemma 2.1. Let $\mathcal{S}$ be a fixed Sylow $p$-subgroup of $\operatorname{Hol}(G,+)$ and $\mathcal{S}^{\prime}$ any other Sylow $p$-subgroup of $\operatorname{Hol}(G,+)$. In the light of first assertion, we only need to observe that any regular subgroup of $\mathcal{S}^{\prime}$ lies in the $\operatorname{Hol}(G,+)$ - conjugacy class of some regular subgroup of $\mathcal{S}$. But this is obvious by Sylow theory. ∎

As a result, we get the following algorithm which improves [15, Algorithm 5.1].

Algorithm 2.3**.**

For a finite group $(G,+)$, the following sequence of computations constructs all non-isomorphic skew left braces $(G,+,\circ):$

1. (1)

Compute $\operatorname{Hol}(G,+)$. 2. (2)

Compute the list of regular subgroups of $\operatorname{Hol}(G)$ of order $|G|$ up to conjugation. 3. (3)

For each representative $\mathcal{G}$ of regular subgroups of $\operatorname{Hol}(G)$, construct the map $\chi:G\to\mathcal{G}$ given by g\mapsto\big{(}f,f(g)^{-1}\big{)}, where \big{(}f,f(g)^{-1}\big{)}\in\mathcal{G}. The triple $(\mathcal{G},G,\chi)$ yields a skew left brace $(G,+,\circ)$ with $\circ:G\times G\to G$ given by g_{1}\circ g_{2}=\chi^{-1}\big{(}\chi(g_{1})\chi(g_{2})\big{)} for all $g_{1},g_{2}\in G$.

As remarked in [15] too, for enumerating skew left braces with additive group $(G,+)$ we only need first two steps of this algorithm.

We also have the following algorithm for finite $p$-groups.

Algorithm 2.4**.**

For a finite $p$-group $(G,+)$, the following sequence of computations constructs all non-isomorphic skew left braces $(G,+,\circ):$

1. (1)

Compute $\operatorname{Hol}(G,+)$. 2. (2)

Compute a representative $\mathcal{S}$ of the conjugacy class of Sylow $p$-subgroups of $\operatorname{Hol}(G,+)$. 3. (3)

Compute the list of regular subgroups of $\mathcal{S}$ of order $|G|$ up to conjugation by the elements of $\operatorname{Aut}(G)$. 4. (4)

For each representative $\mathcal{G}$ of regular subgroups of $\mathcal{S}$ under conjugation action of $\operatorname{Aut}(G)$, construct the map $\chi:G\to\mathcal{G}$ given by g\mapsto\big{(}f,f(g)^{-1}\big{)}, where \big{(}f,f(g)^{-1}\big{)}\in\mathcal{G}. The triple $(\mathcal{G},G,\chi)$ yields a skew left brace $(G,+,\circ)$ with $\circ:G\times G\to G$ given by g_{1}\circ g_{2}=\chi^{-1}\big{(}\chi(g_{1})\chi(g_{2})\big{)} for all $g_{1},g_{2}\in G$.

Notice that for enumerating skew left braces with finite additive $p$-group $(G,+)$ we need first three steps of this algorithm. We conclude this section by reproducing the proof of the following fact.

Proposition 2.5**.**

Let $\mathcal{S}$ be a Sylow $p$-subgroup of $\operatorname{Hol}(G)$ of a finite $p$-group $G$. Then the union of $\operatorname{Aut}(G)$ - conjugacy classes of the regular subgroups of $\mathcal{S}$ constitutes the set of all regular subgroups of $\operatorname{Hol}(G)$.

Proof.

Let $\mathcal{G}$ be an arbitrary regular subgroup of $\operatorname{Hol}(G)$. Then $\mathcal{G}$, being of order $|G|$, is a subgroup of some Sylow $p$-subgroup $\mathcal{S}^{\prime}$ of $\operatorname{Hol}(G)$. By Sylow theory, we know that there exists an element $(\phi,y)\in\operatorname{Hol}(G)$ such that $\mathcal{S}=(\mathcal{S}^{\prime})^{(\phi,y)}$. Thus $\mathcal{G}^{(\phi,y)}$ is a subgroup of $\mathcal{S}$. It follows from the proof of Lemma 2.1 that $\mathcal{G}^{(\phi,y)}=\mathcal{G}^{(\beta,1)}$ for some $\beta\in\operatorname{Aut}(G)$. A routine calculation now shows that $\mathcal{G}^{(\beta,1)}$ is a regular subgroup of $\mathcal{S}$. Indeed, if x^{\big{(}(\psi,z)^{(\beta,1)}\big{)}}=x for some $(\psi,z)\in\mathcal{G}$ and $x\in G$, then it follows that $\beta(x)^{(\psi,z)}=\beta(x)$, which is not possible. This proves that the action of $\mathcal{G}^{(\beta,1)}$ is free on $G$. That the action is transitive, is left as an easy exercise, and the proof is complete. ∎

We remark that on the lines of proof of the preceding proposition, we can easily show that for an arbitrary finite group $G$, $\operatorname{Hol}(G)$ acts on $\operatorname{Reg}(G)$ by conjugation. We have used this fact above without proof as it is well known.

3. Computations

Throughout this section, for a given positive integer $n$, $b(n)$ and $s(n)$, respectively, denote the total number of left braces and skew left braces of order $n$. For each such $n$, $pf(n)$ stands for the prime factorization of $n$. Computations in this section are carried out using Algorithm 2.3. The following table remedies some gaps in the list obtained in [22].

We now enumerate $b(n)$ and $s(n)$ for $n\leq 868$ except some cases for which computations are too big to be handled by our computer. We have given a lower bound on the number of skew left braces of order $3^{5}$, by taking into account the additive groups with Group Id’s $[243,m]$, where $m=1,\ldots,31,33,37,38,40,48,\ldots,63,65,66,67$. By the Group Id we mean the group identification of a group of given order in The Small Groups Library [12] implemented in GAP and MAGMA.

We now record some partial computations considering specific additive groups of given orders.

4. Conclusion and Conjectures

We start by presenting a comparison on the time taken ( in seconds) by [15, Algorithm 5.1] and Algorithm 2.3 for enumerating skew left braces of order 32 for select additive groups which took considerable amount of time on MAGMA.

$\#$ Program was stopped after running more than a month without result.

The data obtained above reveals that Algorithm 2.3 is very expensive, with respect to memory space and time, for handing the situation for prime power orders. So, one really needs to find a substitute for this algorithm. One may think of Algorithm 2.4 as a substitute. But unfortunately, it requires the conjugacy classes of regular subgroups of a given Sylow-$p$-subgroup (of the holomorph of a given finite $p$-group) to be computed in the whole holomorph, which is again very expensive. Although Algorithm 2.4 is not very efficient as such, we hope that it may be improved/modified to handle the computations on skew braces of prime power orders more efficiently.

We now present some conjectures suggested by the big data computed in above tables. It is known from [10] that for a prime integer $q\geq 5$,

[TABLE]

and for prime integers $p$ and $q$ such that $q>p+1>3$,

[TABLE]

For skew left braces, we have

Conjecture 4.1**.**

Let $p$ and $q$ be prime integers. If $q\geq 5$, then

[TABLE]

and if $q>p+1>3$, then

[TABLE]

For prime multiples of $8$ and $12$, we have

Conjecture 4.2**.**

Let $p\geq 11$ be a prime integer. Then

[TABLE]

and

[TABLE]

Conjecture 4.3**.**

Let $p\geq 7$ be a prime integer. Then

[TABLE]

and

[TABLE]

Skew left braces of order $pq$, $p<q$ being prime integers, have been constructed very recently in [1], where it is shown that $s(pq)=1$ if $p\nmid q-1$ and $s(pq)=2p+2$ otherwise. Going a step ahead, we have the following enumeration formula:

Conjecture 4.4**.**

Let $p$ and $q$ be prime integers such that $q>p\geq 3$. Then

[TABLE]

and

[TABLE]

We close with the hope that the readers will be able to use the enormous data produced above to formulate many more conjectures according to their own need and interest.

Acknowledgements. The third named author thanks L. Vendramin for supplying MAGMA codes for computing skew left braces and for his useful comments on the introduction, and acknowledges the support of DST-RSF Grant INT/RUS/RSF/P-2. The first and second named authors acknowledge the support from the RFBR-18-01-0057. The authors thank the referee for suggesting useful modifications.