4548.sagews

Let $G= A_4$ and suppose that $G$ acts on itself by conjugation

(a) Determine the conjugacy classes (orbits) of each element of G. 

(b) Determine all of the stabilizer (isotropy) subgroups for each element of $G$. 

G=PermutationGroup([[(1,2,3)],[(2,3,4)]])

G.order()

12

r=G.conjugacy_classes_representatives()
r

[(), (1,2)(3,4), (1,2,3), (1,2,4)]

for g in r:
    cl=[g]
    for h in G:
        k=h*g*h^(-1)
        if k not in cl:
            cl.append(k) 
    print(cl)

[()] [(1,2)(3,4), (1,4)(2,3), (1,3)(2,4)] [(1,2,3), (1,4,2), (1,3,4), (2,4,3)] [(1,2,4), (1,4,3), (1,3,2), (2,3,4)]

for g in G:
    s=[]
    for h in G:
        if g*h*g^(-1)==h:
            s.append(h)
    print(g,"...",s)

((), '...', [(), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3)]) ((2,3,4), '...', [(), (2,3,4), (2,4,3)]) ((2,4,3), '...', [(), (2,3,4), (2,4,3)]) ((1,2)(3,4), '...', [(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)]) ((1,2,3), '...', [(), (1,2,3), (1,3,2)]) ((1,2,4), '...', [(), (1,2,4), (1,4,2)]) ((1,3,2), '...', [(), (1,2,3), (1,3,2)]) ((1,3,4), '...', [(), (1,3,4), (1,4,3)]) ((1,3)(2,4), '...', [(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)]) ((1,4,2), '...', [(), (1,2,4), (1,4,2)]) ((1,4,3), '...', [(), (1,3,4), (1,4,3)]) ((1,4)(2,3), '...', [(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)])

s=[]
g=G[5]
for h in G:
   if g*h*g^(-1)==h:
       s.append(h)
G5=PermutationGroup(s)
G5

Permutation Group with generators [(), (1,2,4), (1,4,2)]

G5.gens_small()

[(1,4,2)]

G5.gens_small?

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/groups/perm_gps/permgroup.py

Type: <type ‘instancemethod’>

Definition: G5.gens_small()

Docstring:

For this group, returns a generating set which has few elements. As neither irredundancy nor minimal length is proven, it is fast.

EXAMPLES:

sage: R = "(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)" ## R = right
sage: U = "( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)" ## U = top
sage: L = "( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)" ## L = left
sage: F = "(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)" ## F = front
sage: B = "(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)" ## B = back or rear
sage: D = "(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)" ## D = down or bottom
sage: G = PermutationGroup([R,L,U,F,B,D])
sage: len(G.gens_small())
2

The output may be unpredictable, due to the use of randomized algorithms in GAP. Note that both the following answers are equally valid.

sage: G = PermutationGroup([[('a','b')], [('b', 'c')], [('a', 'c')]])
sage: G.gens_small() # random
[('b','c'), ('a','c','b')] ## (on 64-bit Linux)
[('a','b'), ('a','c','b')] ## (on Solaris)
sage: len(G.gens_small()) == 2
True