Coverage for local/lib/python2.7/site-packages/sage/groups/matrix_gps/linear.py : 60%

""" 

Linear Groups 

EXAMPLES:: 

sage: GL(4,QQ) 

General Linear Group of degree 4 over Rational Field 

sage: GL(1,ZZ) 

General Linear Group of degree 1 over Integer Ring 

sage: GL(100,RR) 

General Linear Group of degree 100 over Real Field with 53 bits of precision 

sage: GL(3,GF(49,'a')) 

General Linear Group of degree 3 over Finite Field in a of size 7^2 

sage: SL(2, ZZ) 

Special Linear Group of degree 2 over Integer Ring 

sage: G = SL(2,GF(3)); G 

Special Linear Group of degree 2 over Finite Field of size 3 

sage: G.is_finite() 

True 

sage: G.conjugacy_classes_representatives() 

( 

[1 0] [0 2] [0 1] [2 0] [0 2] [0 1] [0 2] 

[0 1], [1 1], [2 1], [0 2], [1 2], [2 2], [1 0] 

) 

sage: G = SL(6,GF(5)) 

sage: G.gens() 

( 

[2 0 0 0 0 0] [4 0 0 0 0 1] 

[0 3 0 0 0 0] [4 0 0 0 0 0] 

[0 0 1 0 0 0] [0 4 0 0 0 0] 

[0 0 0 1 0 0] [0 0 4 0 0 0] 

[0 0 0 0 1 0] [0 0 0 4 0 0] 

[0 0 0 0 0 1], [0 0 0 0 4 0] 

) 

AUTHORS: 

- William Stein: initial version 

- David Joyner: degree, base_ring, random, order methods; examples 

- David Joyner (2006-05): added center, more examples, renamed random 

attributes, bug fixes. 

- William Stein (2006-12): total rewrite 

- Volker Braun (2013-1) port to new Parent, libGAP, extreme refactoring. 

REFERENCES: See [KL1990]_ and [Car1972]_. 

""" 

#***************************************************************************** 

# Copyright (C) 2006 David Joyner and William Stein <wstein@gmail.com> 

# Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.misc.latex import latex 

from sage.groups.matrix_gps.named_group import ( 

normalize_args_vectorspace, NamedMatrixGroup_generic, NamedMatrixGroup_gap ) 

from sage.categories.groups import Groups 

############################################################################### 

# General Linear Group 

############################################################################### 

def GL(n, R, var='a'): 

""" 

Return the general linear group. 

The general linear group `GL( d, R )` consists of all `d \times d` 

matrices that are invertible over the ring `R`. 

.. NOTE:: 

This group is also available via ``groups.matrix.GL()``. 

INPUT: 

- ``n`` -- a positive integer. 

- ``R`` -- ring or an integer. If an integer is specified, the 

corresponding finite field is used. 

- ``var`` -- variable used to represent generator of the finite 

field, if needed. 

EXAMPLES:: 

sage: G = GL(6,GF(5)) 

sage: G.order() 

11064475422000000000000000 

sage: G.base_ring() 

Finite Field of size 5 

sage: G.category() 

Category of finite groups 

sage: TestSuite(G).run() 

sage: G = GL(6, QQ) 

sage: G.category() 

Category of infinite groups 

sage: TestSuite(G).run() 

Here is the Cayley graph of (relatively small) finite General Linear Group:: 

sage: g = GL(2,3) 

sage: d = g.cayley_graph(); d 

Digraph on 48 vertices 

sage: d.plot(color_by_label=True, vertex_size=0.03, vertex_labels=False) # long time 

Graphics object consisting of 144 graphics primitives 

sage: d.plot3d(color_by_label=True) # long time 

Graphics3d Object 

:: 

sage: F = GF(3); MS = MatrixSpace(F,2,2) 

sage: gens = [MS([[2,0],[0,1]]), MS([[2,1],[2,0]])] 

sage: G = MatrixGroup(gens) 

sage: G.order() 

48 

sage: G.cardinality() 

48 

sage: H = GL(2,F) 

sage: H.order() 

48 

sage: H == G 

True 

sage: H.gens() == G.gens() 

True 

sage: H.as_matrix_group() == H 

True 

sage: H.gens() 

( 

[2 0] [2 1] 

[0 1], [2 0] 

) 

TESTS:: 

sage: groups.matrix.GL(2, 3) 

General Linear Group of degree 2 over Finite Field of size 3 

""" 

degree, ring = normalize_args_vectorspace(n, R, var='a') 

try: 

if ring.is_finite(): 

cat = Groups().Finite() 

else: 

cat = Groups().Infinite() 

except AttributeError: 

cat = Groups() 

name = 'General Linear Group of degree {0} over {1}'.format(degree, ring) 

ltx = 'GL({0}, {1})'.format(degree, latex(ring)) 

try: 

cmd = 'GL({0}, {1})'.format(degree, ring._gap_init_()) 

return LinearMatrixGroup_gap(degree, ring, False, name, ltx, cmd, 

category=cat) 

except ValueError: 

return LinearMatrixGroup_generic(degree, ring, False, name, ltx, 

category=cat) 

############################################################################### 

# Special Linear Group 

############################################################################### 

def SL(n, R, var='a'): 

r""" 

Return the special linear group. 

The special linear group `SL( d, R )` consists of all `d \times d` 

matrices that are invertible over the ring `R` with determinant 

one. 

.. note:: 

This group is also available via ``groups.matrix.SL()``. 

INPUT: 

- ``n`` -- a positive integer. 

- ``R`` -- ring or an integer. If an integer is specified, the 

corresponding finite field is used. 

- ``var`` -- variable used to represent generator of the finite 

field, if needed. 

EXAMPLES:: 

sage: SL(3, GF(2)) 

Special Linear Group of degree 3 over Finite Field of size 2 

sage: G = SL(15, GF(7)); G 

Special Linear Group of degree 15 over Finite Field of size 7 

sage: G.category() 

Category of finite groups 

sage: G.order() 

1956712595698146962015219062429586341124018007182049478916067369638713066737882363393519966343657677430907011270206265834819092046250232049187967718149558134226774650845658791865745408000000 

sage: len(G.gens()) 

2 

sage: G = SL(2, ZZ); G 

Special Linear Group of degree 2 over Integer Ring 

sage: G.category() 

Category of infinite groups 

sage: G.gens() 

( 

[ 0 1] [1 1] 

[-1 0], [0 1] 

) 

Next we compute generators for `\mathrm{SL}_3(\ZZ)` :: 

sage: G = SL(3,ZZ); G 

Special Linear Group of degree 3 over Integer Ring 

sage: G.gens() 

( 

[0 1 0] [ 0 1 0] [1 1 0] 

[0 0 1] [-1 0 0] [0 1 0] 

[1 0 0], [ 0 0 1], [0 0 1] 

) 

sage: TestSuite(G).run() 

TESTS:: 

sage: groups.matrix.SL(2, 3) 

Special Linear Group of degree 2 over Finite Field of size 3 

""" 

degree, ring = normalize_args_vectorspace(n, R, var='a') 

try: 

if ring.is_finite() or n == 1: 

cat = Groups().Finite() 

else: 

cat = Groups().Infinite() 

except AttributeError: 

cat = Groups() 

name = 'Special Linear Group of degree {0} over {1}'.format(degree, ring) 

ltx = 'SL({0}, {1})'.format(degree, latex(ring)) 

from sage.libs.gap.libgap import libgap 

try: 

cmd = 'SL({0}, {1})'.format(degree, ring._gap_init_()) 

return LinearMatrixGroup_gap(degree, ring, True, name, ltx, cmd, 

category=cat) 

except ValueError: 

return LinearMatrixGroup_generic(degree, ring, True, name, ltx, 

category=cat) 

######################################################################## 

# Linear Matrix Group class 

######################################################################## 

class LinearMatrixGroup_generic(NamedMatrixGroup_generic): 

def _check_matrix(self, x, *args): 

"""a 

Check whether the matrix ``x`` is special linear. 

See :meth:`~sage.groups.matrix_gps.matrix_group._check_matrix` 

for details. 

EXAMPLES:: 

sage: G = SL(2,GF(5)) 

sage: G._check_matrix(G.an_element().matrix()) 

""" 

if self._special: 

if x.determinant() != 1: 

raise TypeError('matrix must have determinant one') 

else: 

if x.determinant() == 0: 

raise TypeError('matrix must non-zero determinant') 

class LinearMatrixGroup_gap(NamedMatrixGroup_gap, LinearMatrixGroup_generic): 

pass