abstract algebra

This is a toy tools in hope it's helpful to anyone who's learning abstract algebra :)

example outputs

test_Sn(3)
S(3): [[], [[2, 3]], [[1, 2]], [[1, 2, 3]], [[1, 3, 2]], [[1, 3]]]
Cayley table

['XXX']	[]	[[2, 3]]	[[1, 2]]	[[1, 2, 3]]	[[1, 3, 2]]	[[1, 3]]
[]	[]	[[2, 3]]	[[1, 2]]	[[1, 2, 3]]	[[1, 3, 2]]	[[1, 3]]
[[2, 3]]	[[2, 3]]	[]	[[1, 3, 2]]	[[1, 3]]	[[1, 2]]	[[1, 2, 3]]
[[1, 2]]	[[1, 2]]	[[1, 2, 3]]	[]	[[2, 3]]	[[1, 3]]	[[1, 3, 2]]
[[1, 2, 3]]	[[1, 2, 3]]	[[1, 2]]	[[1, 3]]	[[1, 3, 2]]	[]	[[2, 3]]
[[1, 3, 2]]	[[1, 3, 2]]	[[1, 3]]	[[2, 3]]	[]	[[1, 2, 3]]	[[1, 2]]
[[1, 3]]	[[1, 3]]	[[1, 3, 2]]	[[1, 2, 3]]	[[1, 2]]	[[2, 3]]	[]

test_additive_coset(6)
Z6:[0 1 2 3 4 5]
subgroup generated by <3>:[3, 0]
coset generated by 2·<3>:[5, 2]
coset generated by 4·<3>:[1, 4]

test_multiplicative_coset(7)
U7:[1, 2, 3, 4, 5, 6]
subgroup generated by <2>:[2, 4, 1]
coset generated by 3·<2>:[6, 5, 3]

test_additive_subgroup(6)
Z6:[0 1 2 3 4 5]
<0>[0]
<1>[1, 2, 3, 4, 5, 0]
<2>[2, 4, 0]
<3>[3, 0]
<4>[4, 2, 0]
<5>[5, 4, 3, 2, 1, 0]

test_U(8)
Multiplicative Group of Integers Modulo 8 (U8)
Mod: 8
Set: [1, 3, 5, 7]
Cayley table
X 1 3 5 7
1 1 3 5 7
3 3 1 7 5
5 5 7 1 3
7 7 5 3 1
is_associative = True
unique identity: (1) True
is_closed: True
Latin square property: True

test_cyclic(2,6)
a=2 and N=6 are coprime: False
so, a=2 has modular multiplicative inverse: False
cyclic group <2> Mod: 6
Set: [2, 4]
Cayley table
X 2 4
2 4 2
4 2 4
is_associative = True
unique identity: (4) True
is_closed: True
Latin square property: True

test_cyclic(2,5)
a=2 and N=5 are coprime: True
so, a=2 has modular multiplicative inverse: True
cyclic group <2> Mod: 5
Set: [2, 4, 3, 1]
Cayley table
X 2 4 3 1
2 4 3 1 2
4 3 1 2 4
3 1 2 4 3
1 2 4 3 1
is_associative = True
unique identity: (1) True
is_closed: True
Latin square property: True