#556 ⟨a, b | aba=b, aabb=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

1aba2baa3ba2ba3
11aba2baa3ba2ba3
aaa2ba3a3b1baba2
bbbaa2ba2a3ba31a
a2a2a3ba21ba3abba
bababa2aba3a2ba31
a3a31baaba2a2ba3b
ba2ba2ba31babaa2a3
ba3ba3ba3ba1ba2aa2

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. a4 ⇒ 1 [8]
2. ab ⇒ ba3 [6]
3. b2 ⇒ a2 [7]
# ab:aba=b,aabb=1 a/b - -
aaaa=1
ab=baaa
bb=aa

Same cardinality

33 unique, 1183 total

Σ#PresentationPropertiesDescriptionφ
7188⟨a, b | aab=1, bbbb=1⟩Grp Com 8ℤ8727
7330⟨a, b | aa=1, abba=b⟩Noc 860
8898⟨a, b | aa=a, bbbb=a⟩Noc Com 8ℕ(8 = 4)48
8918⟨a, b | aa=b, bbbb=a⟩Noc Com 8ℕ(8 = 1)34
8919⟨a, b | aa=b, bbbb=b⟩Noc Com 8ℕ(8 = 2)46
8961⟨a, b | aa=a, aba=bb⟩Noc 85
91605⟨a, b | aaa=bb, abb=b⟩Noc Com 8ℕ(8 = 3)55
91606⟨a, b | aaa=bb, bab=a⟩Noc Com 814
91615⟨a, b | aab=aa, baa=b⟩Noc 814
91650⟨a, b | aab=bb, baa=a⟩Noc 815
92206⟨a, b | ab=aa, bbbb=a⟩Noc Com 8ℕ(8 = 5)32
92220⟨a, b | bb=aa, aaab=a⟩Noc Com 822
92247⟨a, b | ab=aa, baa=bb⟩Noc 81
92256⟨a, b | ab=aa, bbb=aa⟩Noc 81
92258⟨a, b | ab=aa, bbb=ba⟩Noc 8
92883⟨a, b | aa=a, abbbb=b⟩Noc 814
93107⟨a, b | ab=a, baaa=bb⟩Noc 89
93123⟨a, b | ab=a, bbaa=bb⟩Noc 810
105191⟨a, b | aab=bb, aaaa=b⟩Noc Com 8ℕ(8 = 6)18
106664⟨a, b | aab=b, aaaa=ba⟩Noc 81
107057⟨a, b | ab=aa, aaaa=bb⟩Noc 87
109380⟨a, b | ab=a, bbbb=baa⟩Noc 82
1112606⟨a, b | abbb=aa, bbbb=a⟩Noc Com 8ℕ(8 = 7)14
1120047⟨a, b | aab=a, bbbb=bbb⟩Noc 8
1120051⟨a, b | aab=b, aaaa=abb⟩Noc Com 81
1124863⟨a, b | ab=a, aaaaaa=bb⟩Noc Com 8
1125112⟨a, b | ab=a, bbbbbb=aa⟩Noc Com 8
1125114⟨a, b | ab=a, bbbbbb=ba⟩Noc 8
1125411⟨a, b | ab=a, aaaaa=bbb⟩Noc Com 8
1125652⟨a, b | ab=a, bbbbb=aaa⟩Noc Com 8
1125658⟨a, b | ab=a, bbbbb=bba⟩Noc 8
1125897⟨a, b | ab=a, bbbb=aaaa⟩Noc Com 8
1125911⟨a, b | ab=a, bbbb=bbba⟩Noc 8

Isomorphic instances

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

28 total

Σ#PresentationMapping
8558⟨a, b | aba=b, abba=1⟩φ(a) = a, φ(b) = b
8560⟨a, b | aba=b, baab=1⟩φ(a) = a, φ(b) = b
103806⟨a, b | aaab=ba, aabb=1⟩φ(a) = a, φ(b) = b
103809⟨a, b | aaab=ba, abba=1⟩φ(a) = a, φ(b) = b
103812⟨a, b | aaab=ba, baab=1⟩φ(a) = a, φ(b) = b
103815⟨a, b | aaab=ba, bbaa=1⟩φ(a) = a, φ(b) = b
103846⟨a, b | aaba=ab, aabb=1⟩φ(a) = a, φ(b) = b
103849⟨a, b | aaba=ab, abba=1⟩φ(a) = a, φ(b) = b
103852⟨a, b | aaba=ab, baab=1⟩φ(a) = a, φ(b) = b
103855⟨a, b | aaba=ab, bbaa=1⟩φ(a) = a, φ(b) = b
103922⟨a, b | abab=aa, abba=1⟩φ(a) = a, φ(b) = b
103925⟨a, b | abab=aa, baab=1⟩φ(a) = a, φ(b) = b
103928⟨a, b | abab=aa, bbaa=1⟩φ(a) = a, φ(b) = b
103989⟨a, b | baba=aa, bbaa=1⟩φ(a) = a, φ(b) = b
105596⟨a, b | aabb=1, aababa=1⟩φ(a) = a, φ(b) = b
105602⟨a, b | aabb=1, abaaab=1⟩φ(a) = a, φ(b) = b
105604⟨a, b | aabb=1, ababaa=1⟩φ(a) = a, φ(b) = b
105613⟨a, b | aabb=1, baaaba=1⟩φ(a) = a, φ(b) = b
105616⟨a, b | aabb=1, babaaa=1⟩φ(a) = a, φ(b) = b
105663⟨a, b | abba=1, aaabab=1⟩φ(a) = a, φ(b) = b
105667⟨a, b | abba=1, aababa=1⟩φ(a) = a, φ(b) = b
105673⟨a, b | abba=1, abaaab=1⟩φ(a) = a, φ(b) = b
105678⟨a, b | abba=1, ababbb=1⟩φ(a) = a, φ(b) = b
105681⟨a, b | abba=1, abbbab=1⟩φ(a) = a, φ(b) = b
105688⟨a, b | abba=1, bababb=1⟩φ(a) = a, φ(b) = b
106207⟨a, b | aba=b, aaabab=1⟩φ(a) = a, φ(b) = b
106211⟨a, b | aba=b, aababa=1⟩φ(a) = a, φ(b) = b
106217⟨a, b | aba=b, abaaab=1⟩φ(a) = a, φ(b) = b