#3134 ⟨a, b | ab=a, bbbb=ba⟩

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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.
A left zero element x satisfies xy = x for all y.
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.

1abbab2b3
11abbab2b3
aaaaaaa
bbbab2bab3ba
bababababababa
b2b2bab3bababa
b3b3bababababa

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. a2 ⇒ a [3]
2. ab ⇒ a [1]
3. b2a ⇒ ba [4]
4. b4 ⇒ ba [2]
# ab:ab=a,bbbb=ba ab - -
aa=a
ab=a
bba=ba
bbbb=ba

Same cardinality

19 unique, 1923 total

Σ#PresentationPropertiesDescriptionφ
622⟨a, b | ab=aa, bb=1⟩Noc 6107
626⟨a, b | aaa=1, abb=1⟩Grp Com 6ℤ61373
7160⟨a, b | ab=aa, bb=b⟩Noc 64
7245⟨a, b | aa=a, bbb=a⟩Noc Com 6ℕ(6 = 3)49
7257⟨a, b | aa=b, bbb=a⟩Noc Com 6ℕ(6 = 1)61
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
8639⟨a, b | ab=aa, baa=b⟩Noc 613
8644⟨a, b | ab=aa, bbb=a⟩Noc Com 6ℕ(6 = 4)46
8893⟨a, b | aa=a, abbb=b⟩Noc 619
81011⟨a, b | ab=a, baa=bb⟩Noc 613
91427⟨a, b | abba=b, baba=1⟩Grp 666
91581⟨a, b | aaa=aa, abb=b⟩Noc 65
91648⟨a, b | aab=bb, abb=a⟩Noc Com 613
91686⟨a, b | abb=aa, bbb=a⟩Noc Com 6ℕ(6 = 5)49
93075⟨a, b | ab=a, aaaa=bb⟩Noc Com 614
93132⟨a, b | ab=a, bbbb=aa⟩Noc Com 65
93193⟨a, b | ab=a, bbb=aaa⟩Noc Com 69
93199⟨a, b | ab=a, bbb=bba⟩Noc 63
1124145⟨a, b | aa=a, abbbbba=b⟩Noc Com 6

Isomorphic instances

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

2 total

Σ#PresentationMapping
109381⟨a, b | ab=a, bbbb=bab⟩φ(a) = a, φ(b) = b
1125908⟨a, b | ab=a, bbbb=babb⟩φ(a) = a, φ(b) = b