Finitely presented groups in Sage
===================================================
#Most of the following computations are actually performed by Gap (see https://www.gap-system.org/)
#Gap has a different syntax, but comes with sage. To start it from the terminal prompt, enter "sage-gap". 


====== Free groups
#Multiple ways of defining them 
F.<a,b,c> = FreeGroup();  F
FreeGroup(3)
FreeGroup('a,b,c')
FreeGroup(3,'t')

#words can also be written down in multiple ways
a*b*c*a
F([1,2,3,1])

x = a^2 * b^(-3) * a^(-2)
x.Tietze

#We can also replace the generators by arbitrary ring elements
g =  a * b / c * b^2
M1 = identity_matrix(2)

M2 = matrix([[1,1],[0,1]])

M3 = matrix([[0,1],[1,0]])
g([M1, M2, M3])
g(1,2,3)

====== Finitely presented groups
G = F / [a^2, b^2, c^2, a*b*c*a*b*c]
#Referencing the elements of G 
G.gen(0) * G.gen(1)
G([1,2,-1])

#Up until now, a, b and c still reference elements of F
a.parent() 
#We can change this though. 
G.inject_variables()
a.parent()

#Elements of G and F are not the same! 
F([1])
G([1])
F([1]) is G([1])
#We can convert them from one parent to the other, though. 
G(a*b/c)
F(G(a*b/c))
#Finitely presented groups are implemented via GAP. If you know what you are doing, you can access the underlying LibGap object. 
G.gap()
#This can be used to call GAP functions which are not yet wrapped in Sage. However, it can also cause you some trouble. 

#As for free abelian groups, we can call syntax to replace the generators with a set of arbitrary ring elements, which satisfy the relations of the group. 

G = FreeGroup(2)
G_ab = G / [G([1, 2, -1, -2])];
a,b = G_ab.gens()
g =  a * b
M1 = matrix([[1,0],[0,2]])

M2 = matrix([[0,1],[1,0]])

g(3, 5)
g(M1, M1)
g(M1, M2)
M1*M2 == M2*M1


##Character tables and Cayley graph
#I
G.character_table()  

F.<a,b> = FreeGroup();  
G = F / [a^2*b^(-2), a*b*a*b^(-1)]
G.character_table() 
G.cayley_table() 
G.cayley_graph()
show(G.cayley_graph())  

#A3
G=AlternatingGroup(3)  
G.cayley_table() 
show(G.cayley_graph())

#A4
G=AlternatingGroup(4)  
G.cayley_table() 
show(G.cayley_graph())
#A5
G=AlternatingGroup(5)  
G.cayley_table() 
show(G.cayley_graph())


Finitely presented groups in GAP
=============================
f := FreeGroup( "a", "b" );;
g:=f/[f.1^2*f.2^(-2),f.1*f.2*f.1*f.2^(-1)];
CharacterTable( g );;
PrintCharacterTable( CharacterTable( g ), "tbl" );