#22283 ⟨a, b | aaa=1, abbbab=ba⟩

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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.

Complete 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
  1. b10 ⇒ b
  2. bab9 ⇒ ba
  3. b9a ⇒ ab9
  4. ab3a ⇒ bab8
  5. ba2b9 ⇒ ba2
  6. bab2a ⇒ ab4ab8
  7. b(ba)2 ⇒ ab5ab7
  8. b3a2 ⇒ (ab6)2
  9. b4ab8a ⇒ ab7ab5
  10. b5ab7a ⇒ ab8ab4
  11. (b6a)2 ⇒ a2b3
  12. b7ab5a ⇒ abab2
  13. b8ab4a ⇒ ab2ab
  14. a3 ⇒ 1
  15. a2ba ⇒ b3ab
  16. a2b2a ⇒ (b3ab)2
  17. a2b6a ⇒ bab8ab3
  18. a2b8a ⇒ b5ab6ab3
  19. aba2 ⇒ b3ab6ab7
  20. abab4a ⇒ b(bab4)2
  21. abab5a ⇒ ba2b2
  22. abab8a ⇒ b6ab6
  23. ab2ab4a ⇒ b(ab)2
  24. ab2ab5a ⇒ b3(ab5)2
  25. ab2ab7a ⇒ b6ab4ab5
  26. a(b4a)2 ⇒ b4(ab5)2
  27. ab4ab6a ⇒ b7ab4ab5
  28. ab5ab6a ⇒ b8ab6
  29. ab(b5a)2 ⇒ b2a2b6
  30. ba2b4a ⇒ a(b5a)2b4
  31. ba2b5a ⇒ abab6ab8
  32. (ba)2b7a ⇒ a2b4ab5
  33. bab4ab7a ⇒ ab2a2b
  34. ba(b5a)2 ⇒ ab2ab6ab3
  35. bab7ab4a ⇒ ab6ab4ab2
  36. b2a2b7a ⇒ ab7ab4ab7
  37. b3ab4ab5a ⇒ (ab)3
  38. b(b2ab4)2a ⇒ ab2ab8ab3
  39. b(b5a)2b4a ⇒ a2b5ab
  40. a2b4ab5a ⇒ b4(ab4)2
  41. a2b4ab7a ⇒ (b5a)2b4ab3
  42. a2b(b4a)2 ⇒ (ba)3b3
  43. a2(b5a)2 ⇒ b3ab5(ab4)2
  44. a2b7ab4a ⇒ b4ab6ab4
  45. a(ba)3 ⇒ b(b4a)2b6
  46. (ab)3b5a ⇒ b(bab4)2ab3
  47. abab6ab4a ⇒ b4ab5(ab4)2
  48. a(b2ab4)2a ⇒ bab6ab4ab7
  49. ab3(bab4)2a ⇒ (b2ab4)2ab7
  50. a(b5a)2b4a ⇒ b3(bab6)2 
# 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

Right Cayley graph

Left Cayley graph