Given an automatic group G return true if G has finite order and false otherwise. If G does have finite order also return the order of G.
The order of the group G as an integer. If the order of G is known to be infinite, the symbol ∞ is returned.
We construct the group Z wreath C2 and compute its order. The result of Infinity indicates that the group has infinite order.
> F<a,b,t> := FreeGroup(3); > Q := quo< F | t^2=1, b*a=a*b, t*a*t=b>; > f, G := IsAutomaticGroup(Q); Running Knuth-Bendix with the following parameter values MaxRelations = 200 MaxStates = 0 TidyInt = 20 MaxWdiffs = 512 HaltingFactor = 100 MinTime = 5 #System is confluent. #Halting with 14 equations. #First word-difference machine with 14 states computed. #Second word-difference machine with 14 states computed. #System is confluent, or halting factor condition holds. #Word-acceptor with 6 states computed. #General multiplier with 27 states computed. #Validity test on general multiplier succeeded. #Checking inverse and short relations. #Axiom checking succeeded. > Order(G); Infinity
> FG<a,b> := FreeGroup(2); > F := quo< FG | a^3=1, b^3=1, (a*b)^4=1, (a*b^-1)^5=1>; > f, G := IsAutomaticGroup(F : GeneratorOrder := [a,b,a^-1,b^-1]); Running Knuth-Bendix with the following parameter values MaxRelations = 200 MaxStates = 0 TidyInt = 20 MaxWdiffs = 512 HaltingFactor = 100 MinTime = 5 #System is confluent. #Halting with 183 equations. #First word-difference machine with 289 states computed. #Second word-difference machine with 360 states computed. #System is confluent, or halting factor condition holds. #Word-acceptor with 314 states computed. #General multiplier with 1638 states computed. #Multiplier incorrect with generator number 4. #General multiplier with 1958 states computed. #Multiplier incorrect with generator number 2. #General multiplier with 2020 states computed. #Multiplier incorrect with generator number 1. #General multiplier with 2038 states computed. #Validity test on general multiplier succeeded. #General length-2 multiplier with 4252 states computed. #Checking inverse and short relations. #Checking relation: _1*_2*_1*_2 = _4*_3*_4*_3 #Checking relation: _1*_4*_1*_4*_1 = _2*_3*_2*_3*_2 #Axiom checking succeeded. > IsFinite(G); true 1080 > isf, ord := IsFinite(G); > isf, ord; true 1080