How to find nilpotent elements of group algebra F2[Q8] with index of nilpotency? Also if the answer is coming in permutation form, how to convert that into elements of F2[Q8] like in the form i,j,k

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asked 0 years ago

Maths is everywhere
3 ●1

Let $\mathbb{F_2}$ be the ring of integers modulo 2 and $\mathcal{Q_8}$ be the group of quaternions such that $\mathcal{Q_8}= {e,\bar{e},i,\bar{i},j,\bar{j},k,\bar{k}}$ . Then $\mathbb{F_2}[\mathcal{Q_8}]$ is the groupring. How can I find its nilpotent elements with index of nilpotency?

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Please always give all information you have on a specific issue, here and elsewhere. If there is already a piece of code or a structural insight, then provide the code, the adjacent mathematical results and/or the links.

dan_fulea ( 0 years ago )

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answered 0 years ago

dan_fulea
5650 ●5 ●44 ●97

There is a related question with an answer providing code on MSE.

I'll use my code posted there also here:

F = GF(2)
Q8 = QuaternionGroup()
A = GroupAlgebra(Q8, F)

elements = [sum(f*A(q) for f, q in zip(f8, list(Q8)))
            for f8 in cartesian_product([F, F, F, F, F, F, F, F])] 

nilpotents = [a for a in elements if a^1024 == A(0)]
nilpotents_with_their_index = []    # and we append
for a in nilpotents:
    for power in [1..1024]:
        if a^power == A(0):
            nilpotents_with_their_index.append((a, power))
            break    # the last loop on power and consider next a, if any more

The code is not optimal, but written so that it can be digested, and the few seconds do not make any difference. We have (after a copy+paste into the sage-ipython-interpreter, followed by a dialog with this code consumer):

sage: len(nilpotents)
128
sage: len(nilpotents_with_their_index)
128
sage: len(elements)
256

We can sort w.r.t. the index, and ask for the maximal index, and how many elements have each index:

sage: nilpotents_with_their_index.sort(key=lambda tup: tup[1])
sage: powers = list(set(tup[1] for tup in nilpotents_with_their_index))
sage: powers
[1, 2, 3, 4]
sage: for power in powers:
....:     print(f"Nilpotency Index = {power} :: "
....:           f"{len([tup for tup in nilpotents_with_their_index if tup[1] == power])}"
....:           f" element(s)")
....: 
Nilpotency Index = 1 :: 1 element(s)
Nilpotency Index = 2 :: 15 element(s)
Nilpotency Index = 3 :: 16 element(s)
Nilpotency Index = 4 :: 96 element(s)

Note that the nilpotent elements are exactly the elements with augmentation zero:

sage: def aug(a):    return sum(a.coefficients() + [A(0)])
sage: set(aug(a) for a in nilpotents)
{0}
sage: set(aug(a) for a in elements if a not in nilpotents)
{()}
sage: # is this last result the set { A(1) }?
sage: _ == set([ A(1) ])
True

link

Comments

Realizing the representation w.r.t. the basis $\pm 1,\pm i,\pm j,\pm k$ is left as an exercise. Just put the hand on some elements that "match" $-1,i,j,k$ , and make a dictionary string to $A$ -element, by using eval. Here $A$ is our group ring. To get the wanted representation, invert the dictionary, or use a function doing this when it is needed, or make a class and define the __str__ method to fit this wanted representation. I am not doing this here.

dan_fulea ( 0 years ago )

F = GF(2) Q8 = QuaternionGroup() A = GroupAlgebra(Q8, F)

elements = [sum(f*A(q) for f, q in zip(f8, list(Q8))) for f8 in cartesian_product([F, F, F, F, F, F, F, F])]

nilpotents = [a for a in elements if a^1024 == A(0)] nilpotents_with_their_index = [] # and we append for a in elements: for b in elements: for c in elements: for d in elements: if ac==0 and ad+bc==0 and bd==0 and a!=0: print(f'{a,b,c,d}')

Maths is everywhere ( 0 years ago )

The above code gives no solution while there is clearly a solution, if we take a nonzero and all others are zero.

Maths is everywhere ( 0 years ago )

This comment is at the wrong place. It comes with no relation to the stated question. Is it so? On the other side, there are a lot of four-tuples $(a,b,c,d)$ that fulfill $a\ne 0$ , $ac=bd=ad+bc=0$ . And there are a lot of solutions.

dan_fulea ( 0 years ago )

For instance:

sage: found = False
sage: for a in elements:
....:     if a == 0 or found:   continue
....:     for c in elements:
....:         if a*c == 0 and not found:
....:             for b in elements:
....:                 if found:    continue
....:                 for d in elements:
....:                     if a*d + b*c == 0 and b*d == 0 and not found:
....:                         print(f'a={a} b={b} c={c} d={d}')
....:                         found = True
....:                         break
....: 
a=(1,6,3,8)(2,5,4,7) b=0 c=0 d=0

dan_fulea ( 0 years ago )

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How to find nilpotent elements of group algebra F2[Q8] with index of nilpotency? Also if the answer is coming in permutation form, how to convert that into elements of F2[Q8] like in the form i,j,k

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How to find nilpotent elements of group algebra F2[Q8] with index of nilpotency? Also if the answer is coming in permutation form, how to convert that into elements of F2[Q8] like in the form i,j,k