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Exercise 2.5.4 (Find all pairs of elements that generate $D_8$)
Use the given lattice to find all pairs of elements that generate (there are pairs).
Answers
Proof. In the lattice of subgroups of we see for instance that the upper limit of and is , so , but . Reasoning with each pair, we obtain the twelve pairs generating :
□With Sagemath:
sage: D8 = DihedralGroup(4); D8.list()
[(), (1,4)(2,3), (1,2,3,4), (1,3)(2,4), (1,3), (2,4), (1,4,3,2), (1,2)(3,4)]
sage: [r,s] = D8.gens(); r, s
((1,2,3,4), (1,4)(2,3))
sage: def my_word_problem(x, n):
....: """ Only for Dihedral Group D_2n"""
....: u, v = -1,-1
....: for i in range(n):
....: for j in range(2):
....: if s^j * r^i == x:
....: u, v = i, j
....: if u == 0:
....: f = ""
....: elif u == 1:
....: f = "r"
....: else:
....: f = "r^{}".format(u)
....: if v == 0:
....: g = ""
....: elif v == 1:
....: g = "s"
....: else:
....: g = "s^{}".format(v)
....: if u==0 and v == 0:
....: f = ’1’
....: return( f + g)
....:
sage: my_word_problem(r * s, 4) # r * s is sr in Dummit & Foote
’r^3s’
sage: for a,b in liste:
....: print(my_word_problem(a, 4),my_word_problem(b, 4))
....:
(’r’, ’s’)
(’r’, ’r^3s’)
(’r^3s’, ’s’)
(’r^3s’, ’r^3’)
(’rs’, ’s’)
(’rs’, ’r’)
(’rs’, ’r^3’)
(’rs’, ’r^2s’)
(’r^3’, ’s’)
(’r^2s’, ’r’)
(’r^2s’, ’r^3s’)
(’r^2s’, ’r^3’)
