I am new to SageMath and am currently exploring how to find all the normal subgroups of a specific group $G$. Here’s a brief description of $G$:

Consider the ring $R = Z/3Z times Z/3Z$, where $Z/3Z$ denotes the field with $3$ elements. We define $SL_4(R)$ as the special linear group of $4 times 4$ matrices with entries from $R$.

The group $G$ is a subgroup of $SL_4(R)$, generated by the following matrices:

begin{bmatrix}
(1,1) & (1,0) & (0,0) & (0,0) \
(0,0) & (1,1) & (0,0) & (0,0) \
(0,0) & (0,0) & (1,1) & (0,1) \
(0,0) & (0,0) & (0,0) & (1,1)
end{bmatrix},

begin{bmatrix}
(1,1) & (0,1) & (0,0) & (0,0) \
(0,0) & (1,1) & (0,0) & (0,0) \
(0,0) & (0,0) & (1,1) & (1,0) \
(0,0) & (0,0) & (0,0) & (1,1)
end{bmatrix},

begin{bmatrix}
(1,1) & (0,0) & (0,0) & (0,0) \
(0,0) & (1,1) & (1,1) & (0,0) \
(0,0) & (0,0) & (1,1) & (0,0) \
(0,0) & (0,0) & (0,0) & (1,1)
end{bmatrix},

begin{bmatrix}
(1,1) & (0,0) & (0,0) & (0,0) \
(1,0) & (1,1) & (0,0) & (0,0) \
(0,0) & (0,0) & (1,1) & (0,0) \
(0,0) & (0,0) & (0,1) & (1,1)
end{bmatrix},

begin{bmatrix}
(1,1) & (0,0) & (0,0) & (0,0) \
(0,1) & (1,1) & (0,0) & (0,0) \
(0,0) & (0,0) & (1,1) & (0,0) \
(0,0) & (0,0) & (1,0) & (1,1)
end{bmatrix},

begin{bmatrix}
(1,1) & (0,0) & (0,0) & (0,0) \
(0,0) & (1,1) & (0,0) & (0,0) \
(0,0) & (1,1) & (1,1) & (0,0) \
(0,0) & (0,0) & (0,0) & (1,1)
end{bmatrix}

I attempted to write SageMath code to find the normal subgroups of GG, but I encountered errors. Here’s the code I used:

# Define the finite field Z/3Z
Z3 = Integers(3)    

# Define the ring R as Cartesian product of Z/3Z and Z/3Z
R = Z3.cartesian_product(Z3)    

# Define the matrices over R
matrices = [    
    matrix(R, [  
        [(1, 1), (1, 0), (0, 0), (0, 0)],    
        [(0, 0), (1, 1), (0, 0), (0, 0)],    
        [(0, 0), (0, 0), (1, 1), (0, 1)],    
        [(0, 0), (0, 0), (0, 0), (1, 1)]    
    ]),    
    matrix(R, [    
        [(1, 1), (0, 1), (0, 0), (0, 0)],    
        [(0, 0), (1, 1), (0, 0), (0, 0)],    
        [(0, 0), (0, 0), (1, 1), (1, 0)],    
        [(0, 0), (0, 0), (0, 0), (1, 1)]    
    ]),    
    matrix(R, [  
        [(1, 1), (0, 0), (0, 0), (0, 0)],  
        [(0, 0), (1, 1), (1, 1), (0, 0)],  
        [(0, 0), (0, 0), (1, 1), (0, 0)],  
        [(0, 0), (0, 0), (0, 0), (1, 1)]  
    ])  
]  

# Include their transposes
matrices += [m.transpose() for m in matrices]

# Generate G as the group generated by these matrices
G = MatrixGroup(matrices)

# Find the normal subgroups of G
normal_subgroups = G.normal_subgroups()

# Display the normal subgroups
normal_subgroups

Expected Outcome: I expected the code to list all the normal subgroups of the group $G$.

Actual Outcome: I encountered the following error message:

---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
File /home/sc_serv/sage/src/sage/structure/category_object.pyx:847, in sage.structure.category_object.CategoryObject.getattr_from_category()
    846 try:
--> 847     return self._cached_methods[name]
    848 except KeyError:

KeyError: 'normal_subgroups'

During handling of the above exception, another exception occurred:

AttributeError                            Traceback (most recent call last)
Cell In [1], line 36
     33 G = MatrixGroup(matrices)
     35 # Find the normal subgroups of G
---> 36 normal_subgroups = G.normal_subgroups()
     38 # Display the normal subgroups
     39 normal_subgroups

File /home/sc_serv/sage/src/sage/structure/category_object.pyx:841, in sage.structure.category_object.CategoryObject.__getattr__()
    839         AttributeError: 'PrimeNumbers_with_category' object has no attribute 'sadfasdf'...
    840     """
--> 841     return self.getattr_from_category(name)
    842 
    843 cdef getattr_from_category(self, name) noexcept:

File /home/sc_serv/sage/src/sage/structure/category_object.pyx:856, in sage.structure.category_object.CategoryObject.getattr_from_category()
    854     cls = self._category.parent_class
    855 
--> 856 attr = getattr_from_other_class(self, cls, name)
    857 self._cached_methods[name] = attr
    858 return attr

File /home/sc_serv/sage/src/sage/cpython/getattr.pyx:357, in sage.cpython.getattr.getattr_from_other_class()
    355     dummy_error_message.cls = type(self)
    356     dummy_error_message.name = name
--> 357     raise AttributeError(dummy_error_message)
    358 cdef PyObject* attr = instance_getattr(cls, name)
    359 if attr is NULL:

AttributeError: 'FinitelyGeneratedMatrixGroup_generic_with_category' object has no attribute 'normal_subgroups'

Request: Could someone please help me understand why this error occurs and how to fix it? Specifically, I would like to know:

1. Why the normal_subgroups method is not available for the matrix group $G$.

2. How to correctly find the normal subgroups of $G$ in SageMath.

Any assistance in debugging this code or providing a correct approach would be greatly appreciated.