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# 14.17 Unitary Groups $GU(n,q)$ and $SU(n,q)$

Module: sage.groups.matrix_gps.unitary

These are $n\times n$ unitary matrices with entries in $GF(q^2)$ .

Author Log:

• David Joyner: initial version (2006-3), modified from
• special_linear (by W. Stein)
• David Joyner (2006-05): minor additions (examples, _latex_,
• __str__, gens, as_matrix_group)

{{{#!python
sage: G = SU(3,GF(5))
sage: G.order()
378000
sage: G
Special Unitary Group of degree 3 over Finite Field of size 5
sage: G._gap_init_()
'SU(3, 5)'
sage: G.random()
[      3 2*a + 1   a + 1]
[2*a + 2       4       a]
[4*a + 2   a + 4       1]
sage: G.base_ring()
Finite Field of size 5
sage: G.field_of_definition()
Finite Field in a of size 5^2

}}}

Module-level Functions

 n, R)

 n, R)

Class: GeneralUnitaryGroup_finite_field

class GeneralUnitaryGroup_finite_field

Class: GeneralUnitaryGroup_generic

class GeneralUnitaryGroup_generic

Functions: as_matrix_group, gens

 self)

{{{#!python
sage: G = GU(4,GF(5))
sage: G.gens()
[[  a   0   0   0]
[  0   1   0   0]
[  0   0   1   0]
[  0   0   0 3*a], [      1       0 4*a + 3       0]
[      1       0       0       0]
[      0 2*a + 4       0       1]
[      0 3*a + 1       0       0]]

}}}

Special Functions: __repr__, __str__, _gap_init_, _latex_

 self)

{{{#!python
sage: G = GU(3,GF(5))
sage: G
General Unitary Group of degree 3 over Finite Field of size 5

}}}

 self)

{{{#!python
sage: G = GU(3,GF(5))
sage: print G
GU(3, GF(5))

}}}

 self)

{{{#!python
sage: G = GU(3,GF(5))
sage: G._latex_()
'GU$(3, 5)$'

}}}

Class: SpecialUnitaryGroup_finite_field

class SpecialUnitaryGroup_finite_field

Class: SpecialUnitaryGroup_generic

class SpecialUnitaryGroup_generic

Functions: as_matrix_group, gens

 self)

{{{#!python
sage: G = SU(4,GF(5))
sage: G.as_matrix_group()
Matrix group over Finite Field in a of size 5^2 with 2 generators:
[[[a, 0, 0, 0], [0, 2*a + 3, 0, 0], [0, 0, 4*a + 1, 0], [0, 0, 0, 3*a]],
[[1, 0, 4*a + 3, 0], [1, 0, 0, 0], [0, 2*a + 4, 0, 1], [0, 3*a + 1, 0, 0]]]

}}}

 self)

{{{#!python
sage: G = SU(4,GF(5))
sage: G.gens()
[[      a       0       0       0]
[      0 2*a + 3       0       0]
[      0       0 4*a + 1       0]
[      0       0       0     3*a],
[      1       0 4*a + 3       0]
[      1       0       0       0]
[      0 2*a + 4       0       1]
[      0 3*a + 1       0       0]]

}}}

Special Functions: __repr__, __str__, _gap_init_, _latex_

 self)

{{{#!python
sage: G = SU(3,GF(5))
sage: G
Special Unitary Group of degree 3 over Finite Field of size 5

}}}

 self)

{{{#!python
sage: G = SU(3,GF(5))
sage: print G
SU(3, GF(5))

}}}

 self)

{{{#!python
sage: G = SU(3,GF(5))
sage: G._latex_()
'SU$(3, 5)$'

}}}

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