#1963 ⟨a, b | aba=b, aaabb=1⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Cayley table

1aba2baa3ba2a4ba3a5ba4ba5
11aba2baa3ba2a4ba3a5ba4ba5
aaa2ba5a3ba4baa5ba21ba3ba4
bbbaa3ba2a4ba3a5ba41ba5aa2
a2a2a3ba4a4ba5a5b1baaba2ba3
bababa2a2ba3a3ba4a4ba5a5b1a
a3a3a4ba3a5ba41ba5aba2baba2
ba2ba2ba3aba4a2ba5a3ba4baa51
a4a4a5ba21ba3aba4a2ba5a3bba
ba3ba3ba41ba5aba2baa3ba2a4a5
a5a51baaba2a2ba3a3ba4a4ba5b
ba4ba4ba5a5b1baaba2a2ba3a3a4
ba5ba5ba4baa5ba21ba3aba4a2a3

Right Cayley graph

Left Cayley graph

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a6 ⇒ 1 [15]
2. ab ⇒ ba5 [14]
3. b2 ⇒ a3 [13]
# ab:aba=b,aaabb=1 a/b - -
aaaaaa=1
ab=baaaaa
bb=aaa

Same cardinality

37 unique, 629 total

Σ#PresentationPropertiesDescriptionφ
7124⟨a, b | aab=a, bbb=1⟩Noc 1261
8458⟨a, b | aaaa=1, abbb=1⟩Grp Com 12ℤ12325
8645⟨a, b | ab=aa, bbb=b⟩Noc 121
91646⟨a, b | aab=bb, aba=a⟩Noc 122
91998⟨a, b | aaa=a, bbbb=a⟩Noc Com 12ℕ(12 = 4)6
92018⟨a, b | aaa=b, bbbb=a⟩Noc Com 12ℕ(12 = 1)27
92019⟨a, b | aaa=b, bbbb=b⟩Noc Com 12ℕ(12 = 3)23
92259⟨a, b | ab=aa, bbb=bb⟩Noc 12
105040⟨a, b | aaa=aa, bbbb=a⟩Noc Com 12ℕ(12 = 8)
105072⟨a, b | aaa=ab, bbbb=a⟩Noc Com 12ℕ(12 = 5)7
105092⟨a, b | aaa=bb, bbbb=a⟩Noc Com 12ℕ(12 = 2)43
105202⟨a, b | aab=bb, abba=a⟩Noc 1241
105330⟨a, b | aaa=ab, bab=bb⟩Noc 121
106597⟨a, b | aaa=b, bbbb=bb⟩Noc Com 12ℕ(12 = 6)4
106660⟨a, b | aab=a, bbbb=ba⟩Noc 121
106661⟨a, b | aab=a, bbbb=bb⟩Noc 121
107105⟨a, b | ab=aa, bbaa=bb⟩Noc 122
1112897⟨a, b | aab=aaa, baaa=b⟩Noc 125
1112910⟨a, b | aab=aaa, bbbb=a⟩Noc Com 12ℕ(12 = 9)2
1115428⟨a, b | aaa=aa, abbbb=b⟩Noc 12
1115996⟨a, b | aaa=ab, aabb=bb⟩Noc 122
1116057⟨a, b | aaa=bb, aabb=ab⟩Noc 124
1116104⟨a, b | aab=aa, abab=bb⟩Noc 12
1116274⟨a, b | aab=bb, aaaa=ab⟩Noc 121
1116275⟨a, b | aab=bb, aaaa=ba⟩Noc 12
1116448⟨a, b | aba=bb, aabb=aa⟩Noc 122
1119773⟨a, b | aba=b, bbbbb=aa⟩Noc Com 12
1119917⟨a, b | aaa=b, bbbb=abb⟩Noc Com 12ℕ(12 = 7)2
1120054⟨a, b | aab=b, aaaa=bba⟩Noc 12
1120686⟨a, b | bb=aa, aaaaab=a⟩Noc Com 1215
1120787⟨a, b | ab=aa, baaaa=bb⟩Noc 127
1121086⟨a, b | bb=aa, aaaa=aba⟩Noc 124
1121092⟨a, b | bb=aa, aaab=aba⟩Noc 123
1121112⟨a, b | bb=aa, abab=aba⟩Noc 12
1124147⟨a, b | aa=a, abbbbbb=b⟩Noc 12
1124991⟨a, b | ab=a, baaaaa=bb⟩Noc 12
1125603⟨a, b | ab=a, bbaaa=bbb⟩Noc 12

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

30 total

Σ#PresentationMapping
91966⟨a, b | aba=b, aabba=1⟩φ(a) = a, φ(b) = b
91974⟨a, b | aba=b, baaab=1⟩φ(a) = a, φ(b) = b
104111⟨a, b | bab=aba, aabb=1⟩φ(a) = baa, φ(b) = b
104113⟨a, b | bab=aba, abba=1⟩φ(a) = baa, φ(b) = b
104601⟨a, b | abba=b, aabbb=1⟩φ(a) = b, φ(b) = a
104606⟨a, b | abba=b, abbba=1⟩φ(a) = b, φ(b) = a
104609⟨a, b | abba=b, baabb=1⟩φ(a) = b, φ(b) = a
105881⟨a, b | aabb=1, ababa=b⟩φ(a) = baa, φ(b) = b
105945⟨a, b | abba=1, ababa=b⟩φ(a) = baa, φ(b) = b
105958⟨a, b | abba=1, babab=a⟩φ(a) = baa, φ(b) = b
1111855⟨a, b | aaba=ab, aaabb=1⟩φ(a) = a, φ(b) = b
1111858⟨a, b | aaba=ab, aabba=1⟩φ(a) = a, φ(b) = b
1111864⟨a, b | aaba=ab, abbaa=1⟩φ(a) = a, φ(b) = b
1111869⟨a, b | aaba=ab, baaab=1⟩φ(a) = a, φ(b) = b
1111876⟨a, b | aaba=ab, bbaaa=1⟩φ(a) = a, φ(b) = b
1112019⟨a, b | abab=aa, aabbb=1⟩φ(a) = b, φ(b) = a
1112026⟨a, b | abab=aa, abbba=1⟩φ(a) = b, φ(b) = a
1112031⟨a, b | abab=aa, baabb=1⟩φ(a) = b, φ(b) = a
1112037⟨a, b | abab=aa, bbaab=1⟩φ(a) = b, φ(b) = a
1112040⟨a, b | abab=aa, bbbaa=1⟩φ(a) = b, φ(b) = a
1115160⟨a, b | aab=ba, ababbb=1⟩φ(a) = aa, φ(b) = b
1115166⟨a, b | aab=ba, abbbab=1⟩φ(a) = aa, φ(b) = b
1115180⟨a, b | aab=ba, bababb=1⟩φ(a) = aa, φ(b) = b
1115183⟨a, b | aab=ba, babbba=1⟩φ(a) = aa, φ(b) = b
1115190⟨a, b | aab=ba, bbabab=1⟩φ(a) = aa, φ(b) = b
1115195⟨a, b | aab=ba, bbbaba=1⟩φ(a) = aa, φ(b) = b
1118621⟨a, b | aba=b, aaaabab=1⟩φ(a) = a, φ(b) = b
1118626⟨a, b | aba=b, aaababa=1⟩φ(a) = a, φ(b) = b
1118635⟨a, b | aba=b, aababaa=1⟩φ(a) = a, φ(b) = b
1118646⟨a, b | aba=b, abaaaab=1⟩φ(a) = a, φ(b) = b