#21046 ⟨a, b | ab=aa, bbbb=bba⟩

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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.

1aba2bab2a3ba2b2ab3a4ba3b2a2ba4
11aba2bab2a3ba2b2ab3a4ba3b2a2ba4
aaa2a2a3a3a3a4a4a4a4a4a4a4a4
bbbab2ba2b2ab3ba3b2a2b2a2b2aba4b2a2b2a2b2a2
a2a2a3a3a4a4a4a4a4a4a4a4a4a4a4
bababa2ba2ba3ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4
b2b2b2ab3b2a2b2a2b2ab2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2
a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4
ba2ba2ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2ab2ab2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2
b3b3b2a2b2ab2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2
a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2b2a2
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4

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. b3a ⇒ b2a2 [4]
3. b4 ⇒ b2a [2]
4. a5 ⇒ a4 [3]
5. b2a3 ⇒ b2a2 [5]
# ab:ab=aa,bbbb=bba ab - -
ab=aa
bbba=bbaa
bbbb=bba
aaaaa=aaaa
bbaaa=bbaa

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
1120846⟨a, b | ab=aa, bbbbb=ba⟩Noc 14
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
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