#12187 ⟨a, b | aaaa=ab, babb=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 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.

1aba2abb2a3ab2b3ab3b4ab4b5ab5b6ab6b7ab7
11aba2abb2a3ab2b3ab3b4ab4b5ab5b6ab6b7ab7
aaa2aba3ab6ab2abab7ab3abab4ab2ab5ab3ab6ab4ab7ab5
bbb6b2b4b7b3b2bb4b2b5b3b6b4b7b5bb6
a2a2a3ab6abab4ab7ab6ab5abab6ab2ab7ab3abab4ab2ab5ab3
ababab6ab2ab4ab7ab3ab2abab4ab2ab5ab3ab6ab4ab7ab5abab6
b2b2b7b3b5bb4b3b2b5b3b6b4b7b5bb6b2b7
a3a3abab4ab6ab2ab5ab4ab3ab6ab4ab7ab5abab6ab2ab7ab3ab
ab2ab2ab7ab3ab5abab4ab3ab2ab5ab3ab6ab4ab7ab5abab6ab2ab7
b3b3bb4b6b2b5b4b3b6b4b7b5bb6b2b7b3b
ab3ab3abab4ab6ab2ab5ab4ab3ab6ab4ab7ab5abab6ab2ab7ab3ab
b4b4b2b5b7b3b6b5b4b7b5bb6b2b7b3bb4b2
ab4ab4ab2ab5ab7ab3ab6ab5ab4ab7ab5abab6ab2ab7ab3abab4ab2
b5b5b3b6bb4b7b6b5bb6b2b7b3bb4b2b5b3
ab5ab5ab3ab6abab4ab7ab6ab5abab6ab2ab7ab3abab4ab2ab5ab3
b6b6b4b7b2b5bb7b6b2b7b3bb4b2b5b3b6b4
ab6ab6ab4ab7ab2ab5abab7ab6ab2ab7ab3abab4ab2ab5ab3ab6ab4
b7b7b5bb3b6b2bb7b3bb4b2b5b3b6b4b7b5
ab7ab7ab5abab3ab6ab2abab7ab3abab4ab2ab5ab3ab6ab4ab7ab5

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. b8 ⇒ b [10]
2. ba ⇒ b6 [9]
3. a2b ⇒ ab6 [11]
4. a4 ⇒ ab [1]
# ab:aaaa=ab,babb=b b/a - -
bbbbbbbb=b
ba=bbbbbb
aab=abbbbbb
aaaa=ab

Same cardinality

17 unique, 102 total

Σ#PresentationPropertiesDescriptionφ
81121⟨a, b | aa=1, abbba=b⟩Noc 1817
93387⟨a, b | aa=1, abbbbb=b⟩Noc 1831
105521⟨a, b | aaab=1, bbbbbb=1⟩Grp Com 18ℤ1833
106732⟨a, b | aba=a, aaab=bb⟩Noc 18
106788⟨a, b | aba=b, baab=aa⟩Noc 18
106795⟨a, b | aba=b, bbbb=aa⟩Noc 18
107039⟨a, b | bb=aa, aaaba=b⟩Noc 182
108910⟨a, b | aa=a, bbbbb=ab⟩Noc 181
1115797⟨a, b | aab=bb, bbbba=a⟩Noc 18
1116042⟨a, b | aaa=ab, bbbb=ab⟩Noc 18
1116043⟨a, b | aaa=ab, bbbb=ba⟩Noc 18
1116313⟨a, b | aab=bb, baba=aa⟩Noc 18
1118758⟨a, b | aaa=a, bbbbbb=a⟩Noc Com 18ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Noc Com 18ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181

Isomorphic instances

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

2 total

Σ#PresentationMapping
1112191⟨a, b | aaaa=ab, bbab=b⟩φ(a) = a, φ(b) = b
1112193⟨a, b | aaaa=ab, bbba=b⟩φ(a) = a, φ(b) = b