# Elements of Affine Groups¶

The class in this module is used to represent the elements of AffineGroup() and its subgroups.

EXAMPLES:

sage: F = AffineGroup(3, QQ)
sage: F([1,2,3,4,5,6,7,8,0], [10,11,12])
[1 2 3]     [10]
x |-> [4 5 6] x + [11]
[7 8 0]     [12]

sage: G = AffineGroup(2, ZZ)
sage: g = G([[1,1],[0,1]], [1,0])
sage: h = G([[1,2],[0,1]], [0,1])
sage: g*h
[1 3]     [2]
x |-> [0 1] x + [1]
sage: h*g
[1 3]     [1]
x |-> [0 1] x + [1]
sage: g*h != h*g
True


AUTHORS:

• Volker Braun
class sage.groups.affine_gps.group_element.AffineGroupElement(parent, A, b=0, convert=True, check=True)

An affine group element.

INPUT:

• A – an invertible matrix, or something defining a matrix if convert==True.
• b– a vector, or something defining a vector if convert==True (default: 0, defining the zero vector).
• parent – the parent affine group.
• convert - bool (default: True). Whether to convert A into the correct matrix space and b into the correct vector space.
• check - bool (default: True). Whether to do some
checks or just accept the input as valid.

As a special case, A can be a matrix obtained from matrix(), that is, one row and one column larger. In that case, the group element defining that matrix is reconstructed.

OUTPUT:

The affine group element $$x \mapsto Ax + b$$

EXAMPLES:

sage: G = AffineGroup(2, GF(3))
sage: g = G.random_element()
sage: type(g)
<class 'sage.groups.affine_gps.affine_group.AffineGroup_with_category.element_class'>
sage: G(g.matrix()) == g
True
sage: G(2)
[2 0]     [0]
x |-> [0 2] x + [0]


Conversion from a matrix and a matrix group element:

sage: M = Matrix(4, 4, [0, 0, -1, 1, 0, -1, 0, 1, -1, 0, 0, 1, 0, 0, 0, 1])
sage: A = AffineGroup(3, ZZ)
sage: A(M)
[ 0  0 -1]     [1]
x |-> [ 0 -1  0] x + [1]
[-1  0  0]     [1]
sage: G = MatrixGroup([M])
sage: A(G.0)
[ 0  0 -1]     [1]
x |-> [ 0 -1  0] x + [1]
[-1  0  0]     [1]

A()

Return the general linear part of an affine group element.

OUTPUT:

The matrix $$A$$ of the affine group element $$Ax + b$$.

EXAMPLES:

sage: G = AffineGroup(3, QQ)
sage: g = G([1,2,3,4,5,6,7,8,0], [10,11,12])
sage: g.A()
[1 2 3]
[4 5 6]
[7 8 0]

b()

Return the translation part of an affine group element.

OUTPUT:

The vector $$b$$ of the affine group element $$Ax + b$$.

EXAMPLES:

sage: G = AffineGroup(3, QQ)
sage: g = G([1,2,3,4,5,6,7,8,0], [10,11,12])
sage: g.b()
(10, 11, 12)

inverse()

Return the inverse group element.

OUTPUT:

Another affine group element.

EXAMPLES:

sage: G = AffineGroup(2, GF(3))
sage: g = G([1,2,3,4], [5,6])
sage: g
[1 2]     [2]
x |-> [0 1] x + [0]
sage: ~g
[1 1]     [1]
x |-> [0 1] x + [0]
sage: g * g.inverse()
[1 0]     [0]
x |-> [0 1] x + [0]
sage: g * g.inverse() == g.inverse() * g == G(1)
True

list()

Return list representation of self.

EXAMPLES:

sage: F = AffineGroup(3, QQ)
sage: g = F([1,2,3,4,5,6,7,8,0], [10,11,12])
sage: g
[1 2 3]     [10]
x |-> [4 5 6] x + [11]
[7 8 0]     [12]
sage: g.matrix()
[ 1  2  3|10]
[ 4  5  6|11]
[ 7  8  0|12]
[--------+--]
[ 0  0  0| 1]
sage: g.list()
[[1, 2, 3, 10], [4, 5, 6, 11], [7, 8, 0, 12], [0, 0, 0, 1]]

matrix()

Return the standard matrix representation of self.

EXAMPLES:

sage: G = AffineGroup(3, GF(7))
sage: g = G([1,2,3,4,5,6,7,8,0], [10,11,12])
sage: g
[1 2 3]     [3]
x |-> [4 5 6] x + [4]
[0 1 0]     [5]
sage: g.matrix()
[1 2 3|3]
[4 5 6|4]
[0 1 0|5]
[-----+-]
[0 0 0|1]
sage: parent(g.matrix())
Full MatrixSpace of 4 by 4 dense matrices over Finite Field of size 7
sage: g.matrix() == matrix(g)
True


Composition of affine group elements equals multiplication of the matrices:

sage: g1 = G.random_element()
sage: g2 = G.random_element()
sage: g1.matrix() * g2.matrix() == (g1*g2).matrix()
True