#330 ⟨a, b | aa=1, abba=b⟩

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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.
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.

Cayley table

Idempotents are shown in bold.

1ababb2ab2b3ab3
11ababb2ab2b3ab3
aa1abbab2b2ab3b3
bbab2b2ab3b3abbab2
ababb2ab2b3ab3babb2
b2b2abb3ab2bab3b2ab
ab2ab2bab3b2abb3ab2b
b3b3ab3babb2ab2b3ab3
ab3ab3b3abbab2b2ab3b3

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. b4 ⇒ b [6]
2. ba ⇒ ab2 [4]
3. a2 ⇒ 1 [1]
# ab:aa=1,abba=b b/a - -
bbbb=b
ba=abb
aa=1

Same cardinality

33 unique, 1151 total

Σ#PresentationPropertiesDescriptionφ
7188⟨a, b | aab=1, bbbb=1⟩Grp Com 8ℤ8727
8556⟨a, b | aba=b, aabb=1⟩Grp 828
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.

60 total

Σ#PresentationMapping
7364⟨a, b | aa=1, aba=bb⟩φ(a) = a, φ(b) = b
7367⟨a, b | aa=1, abb=ba⟩φ(a) = a, φ(b) = b
81115⟨a, b | aa=1, ababa=b⟩φ(a) = a, φ(b) = ab
81185⟨a, b | aa=1, abab=ba⟩φ(a) = a, φ(b) = ab
81250⟨a, b | aa=1, bab=aba⟩φ(a) = a, φ(b) = ab
93349⟨a, b | aa=1, aaabba=b⟩φ(a) = a, φ(b) = b
93369⟨a, b | aa=1, abaaba=b⟩φ(a) = a, φ(b) = b
93482⟨a, b | aa=1, aaaba=bb⟩φ(a) = a, φ(b) = b
93485⟨a, b | aa=1, aaabb=ba⟩φ(a) = a, φ(b) = b
93495⟨a, b | aa=1, aabba=ab⟩φ(a) = a, φ(b) = b
93504⟨a, b | aa=1, abaab=ba⟩φ(a) = a, φ(b) = b
93620⟨a, b | aa=1, aaab=bba⟩φ(a) = a, φ(b) = b
93625⟨a, b | aa=1, aaba=abb⟩φ(a) = a, φ(b) = b
93632⟨a, b | aa=1, aabb=aba⟩φ(a) = a, φ(b) = b
93647⟨a, b | aa=1, abba=aab⟩φ(a) = a, φ(b) = b
93662⟨a, b | aa=1, baab=aba⟩φ(a) = a, φ(b) = b
109677⟨a, b | aa=1, aaababa=b⟩φ(a) = a, φ(b) = ab
109719⟨a, b | aa=1, abaaaba=b⟩φ(a) = a, φ(b) = ab
109949⟨a, b | aa=1, aaabab=ba⟩φ(a) = a, φ(b) = ab
109964⟨a, b | aa=1, aababa=ab⟩φ(a) = a, φ(b) = ab
109988⟨a, b | aa=1, abaaab=ba⟩φ(a) = a, φ(b) = ab
1010219⟨a, b | aa=1, aaaba=bab⟩φ(a) = a, φ(b) = ab
1010238⟨a, b | aa=1, aabab=aba⟩φ(a) = a, φ(b) = ab
1010269⟨a, b | aa=1, ababa=aab⟩φ(a) = a, φ(b) = ab
1010306⟨a, b | aa=1, baaab=aba⟩φ(a) = a, φ(b) = ab
1010483⟨a, b | aa=1, abab=aaba⟩φ(a) = a, φ(b) = ab
1010500⟨a, b | aa=1, baaa=abab⟩φ(a) = a, φ(b) = ab
1126453⟨a, b | aa=1, aaaaabba=b⟩φ(a) = a, φ(b) = b
1126475⟨a, b | aa=1, aaabaaba=b⟩φ(a) = a, φ(b) = b
1126487⟨a, b | aa=1, aaabbaaa=b⟩φ(a) = a, φ(b) = b
1126559⟨a, b | aa=1, abaaaaba=b⟩φ(a) = a, φ(b) = b
1126978⟨a, b | aa=1, aaaaaba=bb⟩φ(a) = a, φ(b) = b
1126981⟨a, b | aa=1, aaaaabb=ba⟩φ(a) = a, φ(b) = b
1126992⟨a, b | aa=1, aaaabba=ab⟩φ(a) = a, φ(b) = b
1127001⟨a, b | aa=1, aaabaaa=bb⟩φ(a) = a, φ(b) = b
1127004⟨a, b | aa=1, aaabaab=ba⟩φ(a) = a, φ(b) = b
1127016⟨a, b | aa=1, aaabbaa=ba⟩φ(a) = a, φ(b) = b
1127035⟨a, b | aa=1, aabaaba=ab⟩φ(a) = a, φ(b) = b
1127086⟨a, b | aa=1, abaaaab=ba⟩φ(a) = a, φ(b) = b
1127508⟨a, b | aa=1, aaaaab=bba⟩φ(a) = a, φ(b) = b
1127513⟨a, b | aa=1, aaaaba=abb⟩φ(a) = a, φ(b) = b
1127520⟨a, b | aa=1, aaaabb=aba⟩φ(a) = a, φ(b) = b
1127532⟨a, b | aa=1, aaabaa=bba⟩φ(a) = a, φ(b) = b
1127543⟨a, b | aa=1, aaabba=aab⟩φ(a) = a, φ(b) = b
1127546⟨a, b | aa=1, aaabba=baa⟩φ(a) = a, φ(b) = b
1127560⟨a, b | aa=1, aabaab=aba⟩φ(a) = a, φ(b) = b
1127584⟨a, b | aa=1, aabbaa=aba⟩φ(a) = a, φ(b) = b
1127621⟨a, b | aa=1, abaaba=aab⟩φ(a) = a, φ(b) = b
1127698⟨a, b | aa=1, baaaab=aba⟩φ(a) = a, φ(b) = b
1128040⟨a, b | aa=1, aaaab=abba⟩φ(a) = a, φ(b) = b
1128053⟨a, b | aa=1, aaaba=aabb⟩φ(a) = a, φ(b) = b
1128059⟨a, b | aa=1, aaaba=baab⟩φ(a) = a, φ(b) = b
1128062⟨a, b | aa=1, aaaba=bbaa⟩φ(a) = a, φ(b) = b
1128068⟨a, b | aa=1, aaabb=aaba⟩φ(a) = a, φ(b) = b
1128074⟨a, b | aa=1, aaabb=baaa⟩φ(a) = a, φ(b) = b
1128087⟨a, b | aa=1, aabaa=abba⟩φ(a) = a, φ(b) = b
1128109⟨a, b | aa=1, aabba=aaab⟩φ(a) = a, φ(b) = b
1128112⟨a, b | aa=1, aabba=abaa⟩φ(a) = a, φ(b) = b
1128142⟨a, b | aa=1, abaab=aaba⟩φ(a) = a, φ(b) = b
1128148⟨a, b | aa=1, abaab=baaa⟩φ(a) = a, φ(b) = b