#22271 ⟨a, b | aaa=1, ababbb=ba⟩

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Quick links

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

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. cb ⇒ bc [5]
2. a3 ⇒ 1 [1]
3. b2c ⇒ b [205]
4. bca ⇒ abc [281]
5. bc2 ⇒ c [206]
6. babc ⇒ ba [14]
7. bac2 ⇒ (ab)2 [610]
8. caba ⇒ ac3 [414]
9. cabc ⇒ ca [41]
10. a2bac ⇒ bab2 [10]
11. abab2 ⇒ bac [7]
12. ba2bc ⇒ ba2 [46]
13. bab3 ⇒ a2ba [4]
14. b5 ⇒ c4 [611]
15. ca2ba ⇒ ab3 [569]
16. ca2bc ⇒ ca2 [183]
17. c2a2c ⇒ ba2ca [247]
18. c3a2 ⇒ a2cab [413]
19. c5 ⇒ b4 [606]
20. a2ca2b ⇒ b3a2 [489]
21. abacab ⇒ b4a [73882]
22. ab3a2 ⇒ ca2b [525]
23. ac(ca)2 ⇒ b(ab)2 [73879]
24. ac4a ⇒ b3ac [73869]
25. ba2cab ⇒ c2a2 [274]
26. (bab)2 ⇒ ab4a [73881]
27. b2a2ca ⇒ ca2c [638]
28. b3a2c ⇒ a2ca2 [284]
29. b4ac ⇒ abaca [73860]
30. (ca2)2 ⇒ b2a2c [609]
31. cab2ac ⇒ ac2ab2 [754]
32. (ca)2b2 ⇒ ba2b2a [73727]
33. (cac)2 ⇒ (ab)2a2 [73847]
34. cac4 ⇒ b2aca2 [73902]
35. c(ca)2b ⇒ a2b2ac [73745]
36. c2(ac)2 ⇒ a2(ba)2 [73622]
37. (c2a)2 ⇒ ac2a2b [73917]
38. c2ac3 ⇒ baca2b [73697]
39. c3ac2 ⇒ aca2b2 [73893]
40. c4ab ⇒ a2b3a [73821]
41. c4ac ⇒ (ba)2ca [73898]
42. a(ab)3 ⇒ c(ca)2 [73401]
43. a2(ba)2c ⇒ (b2a)2 [73644]
44. a2b3ac ⇒ c4a [73853]
45. a2b4a ⇒ bacab [73671]
46. a2cab2a ⇒ c2ab3 [689]
47. a2cac2a ⇒ baba2b [73900]
48. a(ac2)2 ⇒ cab3a [73696]
49. a2c3ac ⇒ ca2b3 [73712]
50. aba(ac)2 ⇒ b2acab [73818]
51. aba2c3 ⇒ b4a2 [73896]
52. (ab)2a2b ⇒ cac2a [73828]
53. a(ba)2ca ⇒ b2(ab)2 [73852]
54. abaca2c ⇒ cab2ab [73764]
55. abacac2 ⇒ b3aba [73895]
56. ab2aba2 ⇒ cab4 [73908]
57. a(b2a)2 ⇒ (ba)2c [73658]
58. ab3ab2 ⇒ ba2c2a [73929]
59. ab4ab ⇒ c3aca [73922]
60. aca2b3 ⇒ c3ac [73890]
61. (aca)2b ⇒ b3aca [73686]
62. (aca)2c ⇒ c2ab2a [73938]
63. aca2c3 ⇒ c3ab2 [73863]
64. acab2a2 ⇒ b2a2b2 [73734]
65. acab3a ⇒ c2ac2 [73911]
66. (ac)2a2b ⇒ b(ba)2c [73755]
67. (ac)2a2c ⇒ b2a2ba [73779]
68. a(ca)3 ⇒ c2a2b2 [73780]
69. ac2ab3 ⇒ cab2a [722]
70. ac3aca ⇒ bacab2 [73787]
71. b(a2b)2 ⇒ ac2ab2a [73789]
72. (ba2)2c ⇒ acac2ab [73809]
73. ba2(ba)2 ⇒ c(ac)2 [73913]
74. ba2b2a2 ⇒ ab2a2c2 [73892]
75. ba2b2ac ⇒ (ca)2b [73867]
76. ba2b3a ⇒ c3ab [73930]
77. ba2cac2 ⇒ a2b2aca [73638]
78. ba2c2a2 ⇒ (ac)3c [73601]
79. ba2c2ab ⇒ ab2a2ba [73795]
80. ba2c2ac ⇒ ab3ab [73889]
81. ba2c3a ⇒ a2b4 [73796]
82. baba2ca ⇒ (ab)2a2c [1908]
83. (ba)3a ⇒ ac2ac [73859]
84. b(ab)3 ⇒ ab3aba [73901]
85. (ba)2ca2 ⇒ (ab2)2 [73897]
86. bab(ac)2 ⇒ a2b(ba)2 [73925]
87. bab2a2c ⇒ acab2ab [73762]
88. baca2b2 ⇒ c2ac2 [73954]
89. b(aca)2 ⇒ cab2ab [73951]
90. baca2c2 ⇒ c2ab3 [73726]
91. bacab2a ⇒ ab(ba)2c [73832]
92. bacab3 ⇒ cac2a2 [73928]
93. bacac3 ⇒ a(ba)3 [356]
94. b(ba2)2 ⇒ ab2(ac)2 [73810]
95. b2a(ab)2 ⇒ (ac)2a2 [73618]
96. b2a2b2a ⇒ acab2 [73827]
97. b2a2b3 ⇒ cac3a [73683]
98. b2a2c2a ⇒ (a2b)2b [73689]
99. b2aba2c ⇒ a2cab3 [73597]
100. b(ba)3 ⇒ a(ab2)2 [73899]
101. b(ba)2ca ⇒ c3ac [73880]
102. (b2a)2a ⇒ ab2(ab)2 [73903]
103. b2aca2b ⇒ cac3 [73684]
104. b2aca2c ⇒ cab4 [73962]
105. b2acab2 ⇒ aba2ca [73934]
106. b2a(ca)2 ⇒ a2b2a2b [73939]
107. b3a2b2 ⇒ ac3ac [73633]
108. b3(ab)2 ⇒ ab(ac)2 [73835]
109. b3abac ⇒ c2aca2 [73904]
110. b(b2a)2 ⇒ (ca)2c2 [73915]
111. b3aca2 ⇒ ac4 [73851]
112. b3(ac)2 ⇒ (aca)2 [73907]
113. b4a2b ⇒ aba2c2 [73857]
114. b4aba ⇒ caca2b [73753]
115. b4ab2 ⇒ acac2a2 [73730]
116. ca2b2a2 ⇒ b(ac)3 [73910]
117. ca2b2ab ⇒ (ba)3c [73876]
118. ca2b2ac ⇒ b(aba)2 [73868]
119. ca2b3a ⇒ b4ab [73961]
120. ca2b4 ⇒ a2c3a [73816]
121. ca2cab2 ⇒ baba2c2 [73933]
122. ca2cac2 ⇒ a(ba2)2 [73716]
123. ca2c2ac ⇒ acac3a [73605]
124. ca2c4 ⇒ a2c3ab [73823]
125. cab2a2c ⇒ a2b2a2b [73958]
126. cab(ba)2 ⇒ acab2ab [73765]
127. c(ab2)2 ⇒ abaca2 [73957]
128. cab3ab ⇒ a2c2ac [73862]
129. cab3ac ⇒ bab2a2b [73945]
130. cab4a ⇒ ab2ab [73657]
131. caca2b2 ⇒ a2b(ba)2 [73754]
132. c(aca)2 ⇒ b2(ac)2 [73839]
133. caca2c2 ⇒ aba2c2a [73918]
134. (ca)3a ⇒ ba(ab)2 [73916]
135. (ca)3b ⇒ bacac2a [73854]
136. c(ac)3 ⇒ a2c2a2b [73936]
137. (ca)2c2a ⇒ a2ba2c2 [73920]
138. cac2a2b ⇒ aca2b2a [73894]
139. cac3ac ⇒ b2a2b2 [73741]
140. c2a2b2a ⇒ (aba)2b [73728]
141. c2a2b3 ⇒ b3aba2 [73914]
142. c2ab2a2 ⇒ ba(ca)2b [73891]
143. c2ab2ab ⇒ (aca)2 [73700]
144. c2ab3a ⇒ ab2a2b2 [73738]
145. c2ab4 ⇒ baca2c [73725]
146. c2aca2b ⇒ b3aba [73948]
147. c2aca2c ⇒ b(ac)2a2 [73793]
148. c3ab2a ⇒ b2acac2 [73872]
149. c3ab3 ⇒ aca2c2 [73921]
150. c3aca2 ⇒ b(ba)2c [73923]
151. a2(ba2)2 ⇒ cac2ab2 [73866]
152. (a2b)2ab ⇒ (b2a)2c [73808]
153. a2ba2ca2 ⇒ ba2c4 [73884]
154. a2ba2c2a ⇒ b3ab2 [73858]
155. a2b2a2ba ⇒ caca2c [73768]
156. (a2b2)2 ⇒ cab2a2 [73942]
157. a2b2(ab)2 ⇒ (ba)2ca [73960]
158. a2b(ba)2c ⇒ caca2b [73840]
159. a2b2aca2 ⇒ b2aba2b [73856]
160. a2b2acab ⇒ ba(ac)2 [73848]
161. a2b3aba ⇒ bacac2 [73919]
162. (a2c)2ca ⇒ cac3ab [73940]
163. a2cab4 ⇒ b2aba2 [73815]
164. a2cac3a ⇒ cab2a2b [73611]
165. a2c2a2b2 ⇒ (ca)3 [73932]
166. a2c2ab2a ⇒ ca(ac)2 [73797]
167. a2c3ab2 ⇒ ca2c3 [73871]
168. aba2b4 ⇒ b(a2c)2 [73747]
169. (ab)2a2c2 ⇒ b3acab [73675]
170. ab(ac)2a2 ⇒ (ba)3c [73959]
171. ab(ac)3 ⇒ ba2b2ab [73877]
172. ab3acab ⇒ ca2c2a2 [812]
173. aca2c2a2 ⇒ baba2c2 [73926]
174. acac2ab2 ⇒ (ba2)2 [73799]
175. acac3ab ⇒ ca2c2a [73937]
176. ba2b(ba)2 ⇒ aca2b2 [73829]
177. ba(ab2)2 ⇒ aba(ca)2 [73878]
178. b(a2c)2a ⇒ ab2acab [73955]
179. b(a2c)2c ⇒ aba2b3 [73664]
180. ba(ac)2a2 ⇒ ab(a2c)2 [73785]
181. ba2(ca)2b ⇒ a2b2a2c2 [73743]
182. ba(ac)3 ⇒ aca2c2ab [73580]
183. ba(ba2)2 ⇒ a2cac2 [73772]
184. b(aba)2b ⇒ ca2b2a [73781]
185. baba2b3 ⇒ a(ba2)2 [73639]
186. baba2c2a ⇒ aca2c2 [73909]
187. bab2a2ba ⇒ a2ba2b3 [8]
188. bab2a2b2 ⇒ cab3a [73947]
189. bab2aca2 ⇒ a(ba)3c [73837]
190. bab2acab ⇒ ca2(ca)2 [73701]
191. bab2(ac)2 ⇒ aba2b2ab [73953]
192. b(ac)3c ⇒ (a2b)2ba [73952]
193. bacac2a2 ⇒ a2b3ab [73956]
194. bacac2ab ⇒ ab2acac2 [3787]
195. b2(aba)2 ⇒ a2b2ac [73587]
196. b2aba2b2 ⇒ a2cac3 [73814]
197. (b2a)2ca ⇒ a(ac)3 [73843]
198. b2acac2a ⇒ a(ca)2b [73873]
199. b3aba2b ⇒ ca2c3a [73935]
200. ca(ac)2a2 ⇒ a2c2ab2 [73790]
201. ca2(ca)2b ⇒ (ac)3c2 [73786]
202. ca(ac)3 ⇒ bab2aca [73836]
203. ca2c2a2b ⇒ a(ab)2a2c [73950]
204. ca2c2ab2 ⇒ ab2a2c3 [73946]
205. ca2c3ab ⇒ a2b2(ac)2 [73931]
206. cab2a2ba ⇒ a2c2ab [73604]
207. cab2a2b2 ⇒ aca2c2a [73822]
208. (ca)2c3a ⇒ aca2c2ab [73885]
209. cac2ab2a ⇒ (a2b)2 [73865]
210. cac3ab2 ⇒ b2a2c4 [73941]
211. (a2b)2bab ⇒ b(ac)3 [73944]
212. a2b2a2c3 ⇒ ba2(ca)2 [73784]
213. a2b2acac2 ⇒ ca2c3a [73949]
214. a(ac)3c2 ⇒ baba2b2a [73943]
215. ab2a2c4 ⇒ ca2c2ab [73887]
216. baba2b2ab ⇒ a(ac)3c [73927]
# ab:aaa=1,ababbb=ba abc bbbbbbbb=c custom:0
cb=bc
aaa=1
bbc=b
bca=abc
bcc=c
babc=ba
bacc=abab
caba=accc
cabc=ca
aabac=babb
ababb=bac
baabc=baa
babbb=aaba
bbbbb=cccc
caaba=abbb
caabc=caa
ccaac=baaca
cccaa=aacab
ccccc=bbbb
aacaab=bbbaa
abacab=bbbba
abbbaa=caab
accaca=babab
acccca=bbbac
baacab=ccaa
babbab=abbbba
bbaaca=caac
bbbaac=aacaa
bbbbac=abaca
caacaa=bbaac
cabbac=accabb
cacabb=baabba
caccac=ababaa
cacccc=bbacaa
ccacab=aabbac
ccacac=aababa
ccacca=accaab
ccaccc=bacaab
cccacc=acaabb
ccccab=aabbba
ccccac=babaca
aababab=ccaca
aababac=bbabba
aabbbac=cccca
aabbbba=bacab
aacabba=ccabbb
aacacca=babaab
aaccacc=cabbba
aacccac=caabbb
abaacac=bbacab
abaaccc=bbbbaa
ababaab=cacca
ababaca=bbabab
abacaac=cabbab
abacacc=bbbaba
abbabaa=cabbbb
abbabba=babac
abbbabb=baacca
abbbbab=cccaca
acaabbb=cccac
acaacab=bbbaca
acaacac=ccabba
acaaccc=cccabb
acabbaa=bbaabb
acabbba=ccacc
acacaab=bbabac
acacaac=bbaaba
acacaca=ccaabb
accabbb=cabba
acccaca=bacabb
baabaab=accabba
baabaac=acaccab
baababa=cacac
baabbaa=abbaacc
baabbac=cacab
baabbba=cccab
baacacc=aabbaca
baaccaa=acacacc
baaccab=abbaaba
baaccac=abbbab
baaccca=aabbbb
babaaca=ababaac
bababaa=accac
bababab=abbbaba
babacaa=abbabb
babacac=aabbaba
babbaac=acabbab
bacaabb=ccacc
bacaaca=cabbab
bacaacc=ccabbb
bacabba=abbabac
bacabbb=caccaa
bacaccc=abababa
bbaabaa=abbacac
bbaabab=acacaa
bbaabba=acabb
bbaabbb=caccca
bbaacca=aabaabb
bbabaac=aacabbb
bbababa=aabbabb
bbabaca=cccac
bbabbaa=abbabab
bbacaab=caccc
bbacaac=cabbbb
bbacabb=abaaca
bbacaca=aabbaab
bbbaabb=acccac
bbbabab=abacac
bbbabac=ccacaa
bbbabba=cacacc
bbbacaa=acccc
bbbacac=acaaca
bbbbaab=abaacc
bbbbaba=cacaab
bbbbabb=acaccaa
caabbaa=bacacac
caabbab=bababac
caabbac=babaaba
caabbba=bbbbab
caabbbb=aaccca
caacabb=babaacc
caacacc=abaabaa
caaccac=acaccca
caacccc=aacccab
cabbaac=aabbaab
cabbaba=acabbab
cabbabb=abacaa
cabbbab=aaccac
cabbbac=babbaab
cabbbba=abbab
cacaabb=aabbaba
cacaaca=bbacac
cacaacc=abaacca
cacacaa=baabab
cacacab=bacacca
cacacac=aaccaab
cacacca=aabaacc
caccaab=acaabba
cacccac=bbaabb
ccaabba=abaabab
ccaabbb=bbbabaa
ccabbaa=bacacab
ccabbab=acaaca
ccabbba=abbaabb
ccabbbb=bacaac
ccacaab=bbbaba
ccacaac=bacacaa
cccabba=bbacacc
cccabbb=acaacc
cccacaa=bbabac
aabaabaa=caccabb
aabaabab=bbabbac
aabaacaa=baacccc
aabaacca=bbbabb
aabbaaba=cacaac
aabbaabb=cabbaa
aabbabab=babaca
aabbabac=cacaab
aabbacaa=bbabaab
aabbacab=baacac
aabbbaba=bacacc
aacaacca=cacccab
aacabbbb=bbabaa
aacaccca=cabbaab
aaccaabb=cacaca
aaccabba=caacac
aacccabb=caaccc
abaabbbb=baacaac
ababaacc=bbbacab
abacacaa=bababac
abacacac=baabbab
abbbacab=caaccaa
acaaccaa=babaacc
acaccabb=baabaa
acacccab=caacca
baabbaba=acaabb
baabbabb=abacaca
baacaaca=abbacab
baacaacc=abaabbb
baacacaa=abaacaac
baacacab=aabbaacc
baacacac=acaaccab
babaabaa=aacacc
babaabab=caabba
babaabbb=abaabaa
babaacca=acaacc
babbaaba=aabaabbb
babbaabb=cabbba
babbacaa=abababac
babbacab=caacaca
babbacac=abaabbab
bacacacc=aabaabba
bacaccaa=aabbbab
bacaccab=abbacacc
bbabaaba=aabbac
bbabaabb=aacaccc
bbabbaca=aacacac
bbacacca=acacab
bbbabaab=caaccca
caacacaa=aaccabb
caacacab=acacaccc
caacacac=babbaca
caaccaab=aababaac
caaccabb=abbaaccc
caacccab=aabbacac
cabbaaba=aaccab
cabbaabb=acaacca
cacaccca=acaaccab
caccabba=aabaab
cacccabb=bbaacccc
aabaabbab=bacacac
aabbaaccc=baacaca
aabbacacc=caaccca
aacacaccc=babaabba
abbaacccc=caaccab
babaabbab=aacacacc

Same cardinality

1 unique, 1 total

Σ#PresentationDescriptionRelated
1110664⟨a, b | aaaa=bbb, abab=1⟩Finite non-Abelian group with 336 elements