#22283 ⟨a, b | aaa=1, abbbab=ba⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system

Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Right Cayley graph

Left Cayley graph

Rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
#RuleProof
1. b10 ⇒ b [67]
2. bab9 ⇒ ba [48]
3. b9a ⇒ ab9 [72]
4. ab3a ⇒ bab8 [49]
5. ba2b9 ⇒ ba2 [54]
6. bab2a ⇒ ab4ab8 [50]
7. b(ba)2 ⇒ ab5ab7 [51]
8. b3a2 ⇒ (ab6)2 [44]
9. b4ab8a ⇒ ab7ab5 [114]
10. b5ab7a ⇒ ab8ab4 [65]
11. (b6a)2 ⇒ a2b3 [75]
12. b7ab5a ⇒ abab2 [84]
13. b8ab4a ⇒ ab2ab [88]
14. a3 ⇒ 1 [1]
15. a2ba ⇒ b3ab [3]
16. a2b2a ⇒ (b3ab)2 [6]
17. a2b6a ⇒ bab8ab3 [80]
18. a2b8a ⇒ b5ab6ab3 [94]
19. aba2 ⇒ b3ab6ab7 [71]
20. abab4a ⇒ b(bab4)2 [87]
21. abab5a ⇒ ba2b2 [19]
22. abab8a ⇒ b6ab6 [92]
23. ab2ab4a ⇒ b(ab)2 [8]
24. ab2ab5a ⇒ b3(ab5)2 [105]
25. ab2ab7a ⇒ b6ab4ab5 [66]
26. a(b4a)2 ⇒ b4(ab5)2 [102]
27. ab4ab6a ⇒ b7ab4ab5 [100]
28. ab5ab6a ⇒ b8ab6 [106]
29. ab(b5a)2 ⇒ b2a2b6 [93]
30. ba2b4a ⇒ a(b5a)2b4 [110]
31. ba2b5a ⇒ abab6ab8 [57]
32. (ba)2b7a ⇒ a2b4ab5 [35]
33. bab4ab7a ⇒ ab2a2b [111]
34. ba(b5a)2 ⇒ ab2ab6ab3 [96]
35. bab7ab4a ⇒ ab6ab4ab2 [117]
36. b2a2b7a ⇒ ab7ab4ab7 [81]
37. b3ab4ab5a ⇒ (ab)3 [13]
38. b(b2ab4)2a ⇒ ab2ab8ab3 [95]
39. b(b5a)2b4a ⇒ a2b5ab [115]
40. a2b4ab5a ⇒ b4(ab4)2 [97]
41. a2b4ab7a ⇒ (b5a)2b4ab3 [99]
42. a2b(b4a)2 ⇒ (ba)3b3 [101]
43. a2(b5a)2 ⇒ b3ab5(ab4)2 [116]
44. a2b7ab4a ⇒ b4ab6ab4 [103]
45. a(ba)3 ⇒ b(b4a)2b6 [83]
46. (ab)3b5a ⇒ b(bab4)2ab3 [112]
47. abab6ab4a ⇒ b4ab5(ab4)2 [104]
48. a(b2ab4)2a ⇒ bab6ab4ab7 [113]
49. ab3(bab4)2a ⇒ (b2ab4)2ab7 [118]
50. a(b5a)2b4a ⇒ b3(bab6)2 [79]
# ab:aaa=1,abbbab=ba b/a
bbbbbbbbbb=b
babbbbbbbbb=ba
bbbbbbbbba=abbbbbbbbb
abbba=babbbbbbbb
baabbbbbbbbb=baa
babba=abbbbabbbbbbbb
bbaba=abbbbbabbbbbbb
bbbaa=abbbbbbabbbbbb
bbbbabbbbbbbba=abbbbbbbabbbbb
bbbbbabbbbbbba=abbbbbbbbabbbb
bbbbbbabbbbbba=aabbb
bbbbbbbabbbbba=ababb
bbbbbbbbabbbba=abbab
aaa=1
aaba=bbbab
aabba=bbbabbbbab
aabbbbbba=babbbbbbbbabbb
aabbbbbbbba=bbbbbabbbbbbabbb
abaa=bbbabbbbbbabbbbbbb
ababbbba=bbabbbbbabbbb
ababbbbba=baabb
ababbbbbbbba=bbbbbbabbbbbb
abbabbbba=babab
abbabbbbba=bbbabbbbbabbbbb
abbabbbbbbba=bbbbbbabbbbabbbbb
abbbbabbbba=bbbbabbbbbabbbbb
abbbbabbbbbba=bbbbbbbabbbbabbbbb
abbbbbabbbbbba=bbbbbbbbabbbbbb
abbbbbbabbbbba=bbaabbbbbb
baabbbba=abbbbbabbbbbabbbb
baabbbbba=ababbbbbbabbbbbbbb
bababbbbbbba=aabbbbabbbbb
babbbbabbbbbbba=abbaab
babbbbbabbbbba=abbabbbbbbabbb
babbbbbbbabbbba=abbbbbbabbbbabb
bbaabbbbbbba=abbbbbbbabbbbabbbbbbb
bbbabbbbabbbbba=ababab
bbbabbbbbbabbbba=abbabbbbbbbbabbb
bbbbbbabbbbbabbbba=aabbbbbab
aabbbbabbbbba=bbbbabbbbabbbb
aabbbbabbbbbbba=bbbbbabbbbbabbbbabbb
aabbbbbabbbba=babababbb
aabbbbbabbbbba=bbbabbbbbabbbbabbbb
aabbbbbbbabbbba=bbbbabbbbbbabbbb
abababa=bbbbbabbbbabbbbbb
abababbbbbba=bbabbbbbabbbbabbb
ababbbbbbabbbba=bbbbabbbbbabbbbabbbb
abbabbbbbbabbbba=babbbbbbabbbbabbbbbbb
abbbbabbbbbabbbba=bbabbbbbbabbbbabbbbbbb
abbbbbabbbbbabbbba=bbbbabbbbbbbabbbbbb