#17659 ⟨a, b | aaaa=1, abbab=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. b9 ⇒ b [48]
2. bab8 ⇒ ba [60]
3. b8a ⇒ ab8 [63]
4. ab2a ⇒ bab7 [61]
5. ba2b8 ⇒ ba2 [66]
6. (ba)2 ⇒ ab3ab7 [62]
7. b2a2 ⇒ ab4ab6 [68]
8. b3ab7a ⇒ (ab5)2 [105]
9. b4ab6a ⇒ ab6ab4 [98]
10. (b5a)2 ⇒ ab7ab3 [77]
11. b6ab4a ⇒ a2b2 [24]
12. b7ab3a ⇒ (ab)2 [42]
13. a2ba ⇒ b(bab2)2 [29]
14. a2b4a ⇒ bab7ab2 [64]
15. abab3a ⇒ ba2b [7]
16. abab5a ⇒ b(b3a)2b5 [83]
17. ab3ab4a ⇒ b5ab3ab5 [85]
18. ba3b8 ⇒ ba3 [69]
19. ba2b3a ⇒ abab4ab7 [75]
20. ba2b5a ⇒ ab6ab3ab5 [95]
21. bab6ab3a ⇒ ab4(ab3)2 [70]
22. b2ab3ab5a ⇒ aba2b [34]
23. b(bab3)2a ⇒ abab6ab2 [79]
24. (b3a)2b6a ⇒ ab5ab3ab4 [54]
25. b3ab4ab5a ⇒ ab(b4a)2b3 [107]
26. b2(bab4)2a ⇒ a2b7ab5 [111]
27. (b4a)3 ⇒ a3b4 [102]
28. b4ab5ab3a ⇒ abab7ab4 [110]
29. b3(b3a)3 ⇒ a2b3ab [21]
30. a4 ⇒ 1 [1]
31. a3b3a ⇒ (b2ab)3 [28]
32. a3b6a ⇒ (bab4)2ab2 [93]
33. a3b7a ⇒ (b3ab)3b [92]
34. a2(b3a)2 ⇒ (b3a)2b4 [49]
35. a2b3ab5a ⇒ b(b3a)3b4 [86]
36. a2b(b4a)2 ⇒ b5(ab4)2 [82]
37. a2b6ab3a ⇒ ba3b6 [88]
38. aba3 ⇒ b2(ab4)3b [76]
39. aba2b7a ⇒ b4(ab4)2 [99]
40. a(bab3)2a ⇒ bab5ab3ab5 [96]
41. aba(b4a)2 ⇒ b5(ab3)3b [84]
42. a(b3a)3 ⇒ (b2ab3)2ab5 [97]
43. ab(b2ab3)2a ⇒ b3(ab6)2 [101]
44. ba3b5a ⇒ aba2b6ab7 [104]
45. bab2(b3a)3 ⇒ ab(b3a)3b2 [109]
46. a3b5ab3a ⇒ b2(bab3)2 [106]
47. a2b2(b3a)3 ⇒ b2ab5(ab4)2 [90]
# 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