# Finite Groups, Abelian Groups#

Sage has some support for computing with permutation groups, finite classical groups (such as \(SU(n,q)\)), 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 [(3,4), (1,2,3)(4,5)]
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()
Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])
sage: G.random_element() # random output
(1,5,3)(2,4)
sage: print(latex(G))
\langle (3,4), (1,2,3)(4,5) \rangle
```

You can also obtain the character table (in LaTeX format) in Sage:

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

Sage also includes classical and matrix groups over finite fields:

```
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_classes_representatives()
(
[1 0] [0 6] [0 4] [6 0] [0 6] [0 4] [0 6] [0 6] [0 6] [4 0]
[0 1], [1 5], [5 5], [0 6], [1 2], [5 2], [1 0], [1 4], [1 3], [0 2],
[5 0]
[0 3]
)
sage: G = Sp(4,GF(7))
sage: G
Symplectic Group of degree 4 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
```

You can also compute using abelian groups (infinite and finite):

```
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
```