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Properties of an Automatic Group

IsFinite(G) : GrpRWS -> BoolElt, RngIntElt
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.
Order(G) : GrpRWS -> RngIntElt
# G : GrpRWS -> RngIntElt
The order of the group G as an integer. If the order of G is known to be infinite, the symbol ∞ is returned.

Example GrpAtc_Order (H75E5)

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

Example GrpAtc_Order-2 (H75E6)

We construct a three fold cover of A6 and check whether it has finite order.

> 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

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012