The Klein four group
Described via 2 2
generators a,b with relations a = e, b = e, ba = ab
|
| e a b ab
_____|____________________________
|
e | e a b ab
|
a | a e ab b
|
b | b ab e a
|
ab | ab b a e
The cyclic group of order 6
Described via the 6
generator a with relation a = e
| 2 3 4 5
| e a a a a a
_____|_________________________________________
|
| 2 3 4 5
e | e a a a a a
|
| 2 3 4 5
a | a a a a a e
|
2 | 2 3 4 5
a | a a a a e a
|
3 | 3 4 5 2
a | a a a e a a
|
4 | 4 5 2 3
a | a a e a a a
|
5 | 5 2 3 4
a | a e a a a a
The symmetric group on three elements
Described via 3 2 -1
generators a,b with relations a = e, b = e, ba = a b
| 2 2
| e a a b ab a b
_____|__________________________________________
|
| 2 2
e | e a a b ab a b
|
| 2 2
a | a a e ab a b b
|
2 | 2 2
a | a e a a b b ab
|
| 2 2
b | b a b ab e a a
|
| 2 2
ab | ab b a b a e a
|
2 | 2 2
a b | a b ab b a a e
The dihedral group of order eight
Described via 4 2 -1
generators a,b with relations a = e, b = e, ba = a b
| 2 3 2 3
| e a a a b ab a b a b
_____|________________________________________________________
|
| 2 3 2 3
e | e a a a b ab a b a b
|
| 2 3 2 3
a | a a a e ab a b a b b
|
2 | 2 3 2 3
a | a a e a a b a b b ab
|
3 | 3 2 3 2
a | a e a a a b b ab a b
|
| 3 2 3 2
b | b a b a b ab e a a a
|
| 3 2 3 2
ab | ab b a b a b a e a a
|
2 | 2 3 2 3
a b | a b ab b a b a a e a
|
3 | 3 2 3 2
a b | a b a b ab b a a a e
The quaternion group (of order eight)
Described via 4 2 2 -1
generators a,b with relations a = e, b = a , ba = a b
| 2 3 2 3
| e a a a b ab a b a b
_____|________________________________________________________
|
| 2 3 2 3
e | e a a a b ab a b a b
|
| 2 3 2 3
a | a a a e ab a b a b b
|
2 | 2 3 2 3
a | a a e a a b a b b ab
|
3 | 3 2 3 2
a | a e a a a b b ab a b
|
| 3 2 2 3
b | b a b a b ab a a e a
|
| 3 2 3 2
ab | ab b a b a b a a a e
|
2 | 2 3 3 2
a b | a b ab b a b e a a a
|
3 | 3 2 3 2
a b | a b a b ab b a e a a