#20846 ⟨a, b | ab=aa, bbbbb=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.
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.

1ababb2ab2b3ab3b4ab4b5b6b7b8
11ababb2ab2b3ab3b4ab4b5b6b7b8
aaababab2ab2ab3ab3ab4ab4ab2ab2ab3ab4ab2
bbb5b2b6b3b7b4b8b5b6b6b7b8b6
ababab2ab2ab3ab3ab4ab4ab2ab2ab3ab3ab4ab2ab3
b2b2b6b3b7b4b8b5b6b6b7b7b8b6b7
ab2ab2ab3ab3ab4ab4ab2ab2ab3ab3ab4ab4ab2ab3ab4
b3b3b7b4b8b5b6b6b7b7b8b8b6b7b8
ab3ab3ab4ab4ab2ab2ab3ab3ab4ab4ab2ab2ab3ab4ab2
b4b4b8b5b6b6b7b7b8b8b6b6b7b8b6
ab4ab4ab2ab2ab3ab3ab4ab4ab2ab2ab3ab3ab4ab2ab3
b5b5b6b6b7b7b8b8b6b6b7b7b8b6b7
b6b6b7b7b8b8b6b6b7b7b8b8b6b7b8
b7b7b8b8b6b6b7b7b8b8b6b6b7b8b6
b8b8b6b6b7b7b8b8b6b6b7b7b8b6b7

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. b9 ⇒ b6 [4]
2. ab5 ⇒ ab2 [3]
3. ba ⇒ b5 [2]
4. a2 ⇒ ab [1]
# ab:ab=aa,bbbbb=ba b/a - -
bbbbbbbbb=bbbbbb
abbbbb=abb
ba=bbbbb
aa=ab

Same cardinality

30 unique, 316 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
91682⟨a, b | aba=bb, bab=a⟩Noc 141
93034⟨a, b | aa=a, bbbb=ab⟩Noc 142
103773⟨a, b | aaaa=bb, abbb=1⟩Grp Com 14ℤ14165
105218⟨a, b | aab=bb, bbba=a⟩Noc 146
105404⟨a, b | aab=bb, aba=aa⟩Noc 141
106718⟨a, b | aab=b, bbba=aa⟩Noc 142
1112183⟨a, b | aaaa=ab, baab=b⟩Noc 143
1112206⟨a, b | aaaa=bb, abbb=a⟩Noc Com 141
1112207⟨a, b | aaaa=bb, abbb=b⟩Noc Com 141
1112441⟨a, b | aabb=aa, baab=b⟩Noc 143
1112499⟨a, b | abab=aa, abba=b⟩Noc 14
1114383⟨a, b | aaaa=b, aabbb=a⟩Noc Com 14ℕ(14 = 1)15
1114384⟨a, b | aaaa=b, aabbb=b⟩Noc Com 14ℕ(14 = 4)5
1115532⟨a, b | aaa=bb, abbbb=b⟩Noc Com 14ℕ(14 = 3)11
1115539⟨a, b | aaa=bb, babbb=a⟩Noc Com 14
1116020⟨a, b | aaa=ab, baab=bb⟩Noc 141
1116079⟨a, b | aaa=bb, bbbb=ab⟩Noc 14
1116293⟨a, b | aab=bb, abab=aa⟩Noc 14
1116470⟨a, b | aba=bb, bbbb=aa⟩Noc 14
1119552⟨a, b | aab=b, abbaa=aa⟩Noc 142
1120844⟨a, b | ab=aa, bbbbb=aa⟩Noc 141
1121023⟨a, b | ab=aa, bbaa=bbb⟩Noc 141
1121040⟨a, b | ab=aa, bbbb=aaa⟩Noc 143
1121044⟨a, b | ab=aa, bbbb=baa⟩Noc 141
1121046⟨a, b | ab=aa, bbbb=bba⟩Noc 14
1121110⟨a, b | bb=aa, abab=aaa⟩Noc 142
1124186⟨a, b | aa=a, bbbbbbb=a⟩Noc Com 14ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Noc Com 14ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Noc 14