#16470 ⟨a, b | aba=bb, bbbb=aa⟩

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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.

1aba2abbaa3abababa4(ba)2a5a6a7
11aba2abbaa3abababa4(ba)2a5a6a7
aaa2aba3a4abaa4a5(ba)2a5a7a6a7a2
bbbaabaa4baba5a5(ba)2a7a6a2a7a2a3
a2a2a3a4a4a5a5a5a6a7a6a2a7a2a3
abababaa5a5(ba)2a6a6a7a2a7a3a2a3a4
babaa4baba5a6(ba)2a6a7a2a7a3a2a3a4
a3a3a4a5a5a6a6a6a7a2a7a3a2a3a4
abaabaa5(ba)2a6a7a7a7a2a3a2a4a3a4a5
babbab(ba)2a7a7a2a2a2a3a4a3a5a4a5a6
a4a4a5a6a6a7a7a7a2a3a2a4a3a4a5
(ba)2(ba)2a7a2a2a3a3a3a4a5a4a6a5a6a7
a5a5a6a7a7a2a2a2a3a4a3a5a4a5a6
a6a6a7a2a2a3a3a3a4a5a4a6a5a6a7
a7a7a2a3a3a4a4a4a5a6a5a7a6a7a2

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. a8 ⇒ a2 [11]
2. ba2 ⇒ a4 [12]
3. a2b ⇒ a4 [6]
4. b2 ⇒ aba [1]
5. (ab)2 ⇒ (ba)2 [3]
# ab:aba=bb,bbbb=aa a/b - -
aaaaaaaa=aa
baa=aaaa
aab=aaaa
bb=aba
abab=baba

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
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
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