#16143 ⟨a, b | aab=aa, 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

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.

1aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
11aba2abb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7
aaa2aba2a2ab2a2ab3a2ab4a2ab5a2ab6a2ab7a2
bbb4b2b7b5b3b6b4b7b5b7b6b7b7b7b7b7
a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2a2
ababab4ab2ab7ab5ab3ab6ab4ab7ab5ab7ab6ab7ab7ab7ab7ab7
b2b2b5b3b7b6b4b7b5b7b6b7b7b7b7b7b7b7
ab2ab2ab5ab3ab7ab6ab4ab7ab5ab7ab6ab7ab7ab7ab7ab7ab7ab7
b3b3b6b4b7b7b5b7b6b7b7b7b7b7b7b7b7b7
ab3ab3ab6ab4ab7ab7ab5ab7ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7
b4b4b7b5b7b7b6b7b7b7b7b7b7b7b7b7b7b7
ab4ab4ab7ab5ab7ab7ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7
b5b5b7b6b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab5ab5ab7ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7
b6b6b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab6ab6ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7
b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7ab7

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 ⇒ b7 [4]
2. ba ⇒ b4 [2]
3. a2b ⇒ a2 [1]
4. a3 ⇒ a2 [3]
# ab:aab=aa,bbbb=ba b/a - -
bbbbbbbb=bbbbbbb
ba=bbbb
aab=aa
aaa=aa

Same cardinality

16 unique, 71 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
106428⟨a, b | aab=b, babbb=a⟩Noc 172
106514⟨a, b | aba=b, baaab=a⟩Noc 17
107152⟨a, b | bb=aa, abab=aa⟩Noc 17
107153⟨a, b | bb=aa, abab=ab⟩Noc 17
107154⟨a, b | bb=aa, abab=ba⟩Noc 17
1113141⟨a, b | aab=aaa, bbb=ab⟩Noc 17
1113142⟨a, b | aab=aaa, bbb=ba⟩Noc 17
1113151⟨a, b | aba=aaa, baa=bb⟩Noc 17
1113156⟨a, b | aba=aaa, bbb=ab⟩Noc 17
1113218⟨a, b | abb=aba, bbb=aa⟩Noc 17
1114395⟨a, b | aaaa=b, abbbb=a⟩Noc Com 17ℕ(17 = 1)2
1114396⟨a, b | aaaa=b, abbbb=b⟩Noc Com 17ℕ(17 = 4)2
1114698⟨a, b | aabb=a, baaaa=b⟩Noc 177
1115488⟨a, b | aaa=ab, babbb=b⟩Noc 173
1116521⟨a, b | aab=bb, bab=aaa⟩Noc 17