#17659 ⟨a, b | aaaa=1, abbab=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. b9 ⇒ b
  2. bab8 ⇒ ba
  3. b8a ⇒ ab8
  4. ab2a ⇒ bab7
  5. ba2b8 ⇒ ba2
  6. (ba)2 ⇒ ab3ab7
  7. b2a2 ⇒ ab4ab6
  8. b3ab7a ⇒ (ab5)2
  9. b4ab6a ⇒ ab6ab4
  10. (b5a)2 ⇒ ab7ab3
  11. b6ab4a ⇒ a2b2
  12. b7ab3a ⇒ (ab)2
  13. a2ba ⇒ b(bab2)2
  14. a2b4a ⇒ bab7ab2
  15. abab3a ⇒ ba2b
  16. abab5a ⇒ b(b3a)2b5
  17. ab3ab4a ⇒ b5ab3ab5
  18. ba3b8 ⇒ ba3
  19. ba2b3a ⇒ abab4ab7
  20. ba2b5a ⇒ ab6ab3ab5
  21. bab6ab3a ⇒ ab4(ab3)2
  22. b2ab3ab5a ⇒ aba2b
  23. b(bab3)2a ⇒ abab6ab2
  24. (b3a)2b6a ⇒ ab5ab3ab4
  25. b3ab4ab5a ⇒ ab(b4a)2b3
  26. b2(bab4)2a ⇒ a2b7ab5
  27. (b4a)3 ⇒ a3b4
  28. b4ab5ab3a ⇒ abab7ab4
  29. b3(b3a)3 ⇒ a2b3ab
  30. a4 ⇒ 1
  31. a3b3a ⇒ (b2ab)3
  32. a3b6a ⇒ (bab4)2ab2
  33. a3b7a ⇒ (b3ab)3b
  34. a2(b3a)2 ⇒ (b3a)2b4
  35. a2b3ab5a ⇒ b(b3a)3b4
  36. a2b(b4a)2 ⇒ b5(ab4)2
  37. a2b6ab3a ⇒ ba3b6
  38. aba3 ⇒ b2(ab4)3b
  39. aba2b7a ⇒ b4(ab4)2
  40. a(bab3)2a ⇒ bab5ab3ab5
  41. aba(b4a)2 ⇒ b5(ab3)3b
  42. a(b3a)3 ⇒ (b2ab3)2ab5
  43. ab(b2ab3)2a ⇒ b3(ab6)2
  44. ba3b5a ⇒ aba2b6ab7
  45. bab2(b3a)3 ⇒ ab(b3a)3b2
  46. a3b5ab3a ⇒ b2(bab3)2
  47. a2b2(b3a)3 ⇒ b2ab5(ab4)2 
# ab:aaaa=1,abbab=ba b/a
bbbbbbbbb=b
babbbbbbbb=ba
bbbbbbbba=abbbbbbbb
abba=babbbbbbb
baabbbbbbbb=baa
baba=abbbabbbbbbb
bbaa=abbbbabbbbbb
bbbabbbbbbba=abbbbbabbbbb
bbbbabbbbbba=abbbbbbabbbb
bbbbbabbbbba=abbbbbbbabbb
bbbbbbabbbba=aabb
bbbbbbbabbba=abab
aaba=bbabbbabb
aabbbba=babbbbbbbabb
ababbba=baab
ababbbbba=bbbbabbbabbbbb
abbbabbbba=bbbbbabbbabbbbb
baaabbbbbbbb=baaa
baabbba=ababbbbabbbbbbb
baabbbbba=abbbbbbabbbabbbbb
babbbbbbabbba=abbbbabbbabbb
bbabbbabbbbba=abaab
bbabbbbabbba=ababbbbbbabb
bbbabbbabbbbbba=abbbbbabbbabbbb
bbbabbbbabbbbba=abbbbbabbbbabbb
bbbabbbbbabbbba=aabbbbbbbabbbbb
bbbbabbbbabbbba=aaabbbb
bbbbabbbbbabbba=ababbbbbbbabbbb
bbbbbbabbbabbba=aabbbab
aaaa=1
aaabbba=bbabbbabbbab
aaabbbbbba=babbbbbabbbbabb
aaabbbbbbba=bbbabbbbabbbbabb
aabbbabbba=bbbabbbabbbb
aabbbabbbbba=bbbbabbbabbbabbbb
aabbbbbabbbba=bbbbbabbbbabbbb
aabbbbbbabbba=baaabbbbbb
abaaa=bbabbbbabbbbabbbbb
abaabbbbbbba=bbbbabbbbabbbb
ababbbbabbba=babbbbbabbbabbbbb
ababbbbabbbba=bbbbbabbbabbbabbbb
abbbabbbabbba=bbabbbbbabbbabbbbb
abbbabbbbbabbba=bbbabbbbbbabbbbbb
baaabbbbba=abaabbbbbbabbbbbbb
babbbbbabbbabbba=abbbbabbbabbbabb
aaabbbbbabbba=bbbabbbbabbb
aabbbbbabbbabbba=bbabbbbbabbbbabbbb

Right Cayley graph

Left Cayley graph