#2231 ⟨a, b | ab=aa, aaa=bb⟩

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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 zero element z satisfies zy = yz = z 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.

1aba2bab2ba2
11aba2bab2ba2
aaa2a2b2b2b2b2
bbbab2ba2b2b2b2
a2a2b2b2b2b2b2b2
bababa2ba2b2b2b2b2
b2b2b2b2b2b2b2b2
ba2ba2b2b2b2b2b2b2

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. ab ⇒ a2 [1]
2. a3 ⇒ b2 [2]
3. b2a ⇒ b2 [4]
4. b3 ⇒ b2 [5]
# ab:ab=aa,aaa=bb ab - -
ab=aa
aaa=bb
bba=bb
bbb=bb

Same cardinality

22 unique, 1220 total

Σ#PresentationPropertiesDescriptionφ
7116⟨a, b | aaa=b, abb=1⟩Grp Com 7ℤ7763
8577⟨a, b | aaa=b, abb=a⟩Noc Com 7ℕ(7 = 1)59
8578⟨a, b | aaa=b, abb=b⟩Noc Com 7ℕ(7 = 3)44
8913⟨a, b | aa=b, abbb=b⟩Noc Com 7ℕ(7 = 2)82
8937⟨a, b | ab=a, baaa=b⟩Noc 766
91593⟨a, b | aaa=ab, baa=b⟩Noc 73
91601⟨a, b | aaa=bb, aab=b⟩Noc Com 75
91620⟨a, b | aab=aa, bbb=a⟩Noc Com 7ℕ(7 = 6)35
91632⟨a, b | aab=ab, bbb=a⟩Noc Com 7ℕ(7 = 4)50
92074⟨a, b | aab=b, babb=a⟩Noc Com 720
92271⟨a, b | bb=aa, aaa=ab⟩Noc 73
93197⟨a, b | ab=a, bbb=baa⟩Noc 75
104162⟨a, b | abb=aab, bbb=a⟩Noc Com 7ℕ(7 = 5)42
106251⟨a, b | aaa=a, aabba=b⟩Noc 71
108987⟨a, b | ab=a, aaaaa=bb⟩Noc Com 75
109108⟨a, b | ab=a, bbbbb=aa⟩Noc Com 72
109110⟨a, b | ab=a, bbbbb=ba⟩Noc 71
109263⟨a, b | ab=a, aaaa=bbb⟩Noc Com 74
109375⟨a, b | ab=a, bbba=bbb⟩Noc 73
109376⟨a, b | ab=a, bbbb=aaa⟩Noc Com 73
109382⟨a, b | ab=a, bbbb=bba⟩Noc 71
1114843⟨a, b | abba=b, aabab=a⟩Noc 71

Isomorphic instances

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

3 total

Σ#PresentationMapping
92235⟨a, b | ab=aa, aab=bb⟩φ(a) = a, φ(b) = b
92239⟨a, b | ab=aa, aba=bb⟩φ(a) = a, φ(b) = b
92243⟨a, b | ab=aa, abb=bb⟩φ(a) = a, φ(b) = b