2.7 Finite Groups

Sage has some support for computing with permutation groups, finite classical groups (such as

), finite matrix groups (with your own generators), and abelian groups (even infinite ones). Much of this is implemented using the interface to GAP.

For example, to create a permutation group, give a list of generators, as in the following example.

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(3,4)'])
sage: G
Permutation Group with generators [(1,2,3)(4,5), (3,4)]
sage: G.order()
120
sage: G.is_abelian()
False
sage: G.derived_series()           # random-ish output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)],
 Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (1,3,5)]]
sage: G.center()
Permutation Group with generators [()]
sage: G.random_element()           # random output
(1,5,3)(2,4)
sage: print latex(G)
\langle (1,2,3)(4,5), (3,4) \rangle

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: latex(G.character_table())
\left(\begin{array}{rrrr}
1 & 1 & 1 & 1 \\
1 & 1 & -\zeta_{3} - 1 & \zeta_{3} \\
1 & 1 & \zeta_{3} & -\zeta_{3} - 1 \\
3 & -1 & 0 & 0
\end{array}\right)

sage: MS = MatrixSpace(GF(7), 2)
sage: gens = [MS([[1,0],[-1,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.conjugacy_class_representatives()
    [
    [1 0]
    [0 1],
    [0 1]
    [6 1],
    ...
    [6 0]
    [0 6]
    ]
sage: G = Sp(4,GF(7))
sage: G._gap_init_()
'Sp(4, 7)'
sage: G
Symplectic Group of rank 2 over Finite Field of size 7
sage: G.random_element()             # random output
[5 5 5 1]
[0 2 6 3]
[5 0 1 0]
[4 6 3 4]
sage: G.order()
276595200

sage: F = AbelianGroup(5, [5,5,7,8,9], names='abcde')
sage: (a, b, c, d, e) = F.gens()
sage: d * b**2 * c**3 
b^2*c^3*d
sage: F = AbelianGroup(3,[2]*3); F
Multiplicative Abelian Group isomorphic to C2 x C2 x C2
sage: H = AbelianGroup([2,3], names="xy"); H
Multiplicative Abelian Group isomorphic to C2 x C3
sage: AbelianGroup(5)
Multiplicative Abelian Group isomorphic to Z x Z x Z x Z x Z
sage: AbelianGroup(5).order()
+Infinity