Commutator subgroup

cf. Sage Math docs: Groups



In [30]:

    
G = SL(2,ZZ)



In [31]:

    
G









    Out[31]:





Special Linear Group of degree 2 over Integer Ring



In [32]:

    
identity = matrix(ZZ,[[1,0],[0,1]])



In [33]:

    
G.cayley_table(elements=[identity,-identity])









    Out[33]:





*  a b
 +----
a| a b
b| b a



In [34]:

    
from sage.matrix.operation_table import OperationTable



In [35]:

    
G=DiCyclicGroup(3)



In [36]:

    
commutator = lambda x, y: x*y*x^-1*y^-1



In [37]:

    
T=OperationTable(G,commutator)



In [38]:

    
T









    Out[38]:





.  a b c d e f g h i j k l
 +------------------------
a| a a a a a a a a a a a a
b| a a h d a d h h a h d d
c| a d a a a d d a d d d a
d| a h a a a h h a h h h a
e| a a a a a a a a a a a a
f| a h h d a a d h h d a d
g| a d h d a h a h d a h d
h| a d a a a d d a d d d a
i| a a h d a d h h a h d d
j| a d h d a h a h d a h d
k| a h h d a a d h h d a d
l| a h a a a h h a h h h a



In [39]:

    
trans = T.translation()



In [40]:

    
comm = [trans['a'],trans['d'],trans['h']]



In [41]:

    
comm









    Out[41]:





[(), (5,7,6), (5,6,7)]



In [42]:

    
P=G.cayley_table(elements=comm)



In [43]:

    
P









    Out[43]:





*  a b c
 +------
a| a b c
b| b c a
c| c a b

Finitely Presented Groups

cf. Sage Math documentation, Groups, Finitely Presented Groups in combination with

Joseph J. Rotman. Advanced Modern Algebra (Graduate Studies in Mathematics) 2nd Edition. American Mathematical Society

EY : 20160411 I used the 1st edition of Rotman



In [1]:

    
F.<a,b,c> = FreeGroup()



In [2]:

    
G = F/[a^2,b^2,c^2,a*b*c*a*b*c]



In [3]:

    
G









    Out[3]:





Finitely presented group < a, b, c | a^2, b^2, c^2, (a*b*c)^2 >



In [4]:

    
G.gen(0) * G.gen(1)









    Out[4]:





a*b



In [5]:

    
G([1,2,-1])









    Out[5]:





a*b*a^-1



In [6]:

    
a.parent()









    Out[6]:





Free Group on generators {a, b, c}



In [7]:

    
G.inject_variables()









    



Defining a, b, c



In [8]:

    
a.parent()









    Out[8]:





Finitely presented group < a, b, c | a^2, b^2, c^2, (a*b*c)^2 >

Notice that, even if they are represented in the same way, the elements of a finitely presented group and the elements of the corresponding free group are not the same thing. However, they can be converted from one parent to the other:



In [9]:

    
F.<a,b,c> = FreeGroup()



In [10]:

    
G = F/[a^2,b^2,c^2,a*b*c*a*b*c]



In [11]:

    
F([1])









    Out[11]:





a



In [12]:

    
G([1])









    Out[12]:





a



In [13]:

    
F([1]) is G([1])









    Out[13]:





False



In [14]:

    
F([1]) == G([1])









    Out[14]:





False



In [15]:

    
G(a*b*c)









    Out[15]:





a*b*c



In [16]:

    
F(G(a*b/c))









    Out[16]:





a*b*c^-1

Finitely presented groups are implemented via GAP. You can use the gap() method to access the underlying LibGAP object:



In [17]:

    
G = FreeGroup(2)



In [18]:

    
G.inject_variables()









    



Defining x0, x1



In [19]:

    
H = G/(x0^2,(x0*x1)^2,x1^2)



In [20]:

    
H.gap()









    Out[20]:





<fp group on the generators [ x0, x1 ]>

This can be useful, for example, to use GAP functions that are not yet wrapped in Sage:



In [22]:

    
H.gap().LowerCentralSeries()









    Out[22]:





[ Group(<fp, no generators known>), Group(<fp, no generators known>) ]



In [23]:

    
G = FreeGroup(2)



In [24]:

    
H=G/(G([1,1]),G([2,2,2]),G([1,2,-1,-2])); H









    Out[24]:





Finitely presented group < x0, x1 | x0^2, x1^3, x0*x1*x0^-1*x1^-1 >



In [25]:

    
a = H([1])



In [26]:

    
a









    Out[26]:





x0



In [28]:

    
a.gap()









    Out[28]:





x0



In [45]:

    
a.gap().Order()









    Out[45]:





2



In [46]:

    
type(_) # note that the above output is not a Sage integer









    Out[46]:





<type 'sage.libs.gap.element.GapElement_Integer'>

cf. Rotman, Chapter 5, Groups II Proposition 5.80

generalized quaternion group $\mathbf{Q}_n$, $\forall \, n \geq 3$



In [49]:

    
def make_Q_n(n):
    assert n >= 3
    F.<a,b> = FreeGroup()
    G = F/[a^(2^(n-1)),a*b*a*b^(-1),a^(2^(n-2))*b^(-2)]
    return G



In [50]:

    
Q_3 = make_Q_n(3)



In [51]:

    
Q_3









    Out[51]:





Finitely presented group < a, b | a^4, a*b*a*b^-1, a^2*b^-2 >



In [52]:

    
Q_3.monoid_generators()









    Out[52]:





Family (a, b, a^-1, b^-1)

dihedral group $D_{2n}$



In [53]:

    
def make_D_2n(n):
    assert n > 0
    F.<a,b> = FreeGroup()
    G = F/[a^n,b^2,b*a*b*a]
    return G



In [54]:

    
D_4 = make_D_2n(2)



In [55]:

    
D_4









    Out[55]:





Finitely presented group < a, b | a^2, b^2, (b*a)^2 >

There a large number of modules to explore (by doing the command dir(D_4) or dir() on your group of choice) that I have yet to fully and thoroughly understand: I'd invite you to help me out and add your own explanations and explorations.



In [56]:

    
D_4.abelian_invariants()









    Out[56]:





(2, 2)



In [57]:

    
D_4.alexander_matrix()









    Out[57]:





[    B[1] + B[a]               0]
[              0     B[1] + B[b]]
[B[b] + B[b*a*b]   B[1] + B[b*a]]



In [63]:

    
D_4.center()









    Out[63]:





Group([ a, b ])



In [ ]: