#11515 ⟨a, b | ababa=b, abbaa=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Cayley table

1aba2abbaa3a2bababa2a4a2baaba2ba3a5a2ba2aba3ba4a2ba3aba4ba5a2ba4aba5a2ba5
11aba2abbaa3a2bababa2a4a2baaba2ba3a5a2ba2aba3ba4a2ba3aba4ba5a2ba4aba5a2ba5
aaa2aba3a2babaa4ba3a2baaba2a5ba4a2ba2aba31ba5a2ba3aba4ba2ba4aba5baa2ba5ba2
bbbaa3ba2a2ba2a4ba3abaa2ba3a5ba4aba2a2ba41ba5aba3a2ba5aaba4a2ba2aba5a2baab
a2a2a3a2ba4ba3a2baa5aba3ba4a2ba21aba4ba5a2ba3aaba5ba2ba4abbaa2ba5ababa2aba2
abababaa4aba2ba5a5aba3a2bab1aba4a2ba2baaaba5a2ba3ba2a2a2ba4ba3a3a2ba5ba4a2b
bababa2a2ba2ba3abaa2ba3ba41aba2a2ba4ba5aaba3a2ba5ba2aba4a2ba3aba5a2baa4aba5
a3a3a4ba3a5aba3ba41a2ba3aba4ba5aa2ba4aba5ba2a2ba5abbaa2bababa2a2baaba2a2ba2
a2ba2ba2baa5a2ba2aba51a2ba3ba4abaa2ba4ba5abaa2a2ba5baba2a3baaba3a4ba2aba4ba3
abaabaaba2ba5aba3a2bababa4aa2ba2baaba5a2a2ba3ba2aba3a2ba4ba3a4a2ba5ba4a5a2b1
ba2ba2ba3ababa41aba2ba5a2ba5aaba3ba2ba2aba4baa2baa3aba5a2ba2a4aba2ba3a5a2ba4
a4a4a5aba31a2ba3aba4aba2ba4aba5a2baa2ba5aba3ba2a2bababa3a2baaba2ba4a2ba2ba5
a2baa2baa2ba2aba5a2ba3ba4aba2ba4a2ba5abaa2ba5a3baba2a2ba4baaba3a5ba2aba41ba3a
aba2aba2aba3a2baaba4aa2ba2aba5ba2a2a2ba3abba3a3a2ba4ababa4a4a2ba5ba5a5a2bb1ba
ba3ba3ba41ba5a2ba5ababa4a2ba2baaba5a2baa3ba2aba2ba2a4abaa2ba3a5aba2a2ba4aba3
a5a51a2ba3aba2ba4a2abbaa2ba5a3ababa2a2ba4aba2ba3a2baaba3ba4a2ba2aba4ba5aba5
a2ba2a2ba2a2ba3ba4a2ba4a2ba5a2ba5aba2a3ba2baba3a4baa2baaba4a5ba2aba51ba3abaaba
aba3aba3aba4aaba5ba2a2aba2ba4ba3a3abaa2ba5ba4a4aba2a2bba5a5a2bab1a2ba2baa2ba3
ba4ba4ba5a2ba5baba4a2bbaa3aba5a2baba2a4aba2ba2ba3a5abaa2ba31aba2a2ba4aaba3a2
a2ba3a2ba3a2ba4a2a2ba5aba2a3a2bbaaba3a4a2baba2aba4a5a2ba2ba3aba51ba4ababa5abab
aba4aba4aba5ba2aba2ba4ba3abaa4a2ba5ba4aba2a5a2bba5aba31a2babaa2ba2baa2a2ba3a3
ba5ba5baba4baa3aba5ba2a2ba2a4abba3a2ba3a5ababa4a2ba41aba2a2ba5aaba3a2ba2a2ba
a2ba4a2ba4a2ba5aba2a2bbaaba3a2baa5ba2aba4a2ba21ba3aba5a2ba3aba4aba2ba5abaa3ba4
aba5aba5aba2ba4abaa4a2ba5aba2ba5a5a2baba3b1a2baaba4baaa2ba2ba2a2a2ba3ba3a3ba4
a2ba5a2ba5a2bbaa2baa5ba2a2ba2aba51ba3a2ba3ababa4a2ba4abaa2ba5aba2a3baba3a4aba4

Right Cayley graph

Left Cayley graph

Rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
#RuleProof
1. a6 ⇒ 1 [20]
2. a3b ⇒ ba3 [21]
3. b2 ⇒ a3 [19]
4. bab ⇒ a2ba2 [14]
5. ba2b ⇒ aba [13]
# ab:ababa=b,abbaa=1 a/b
aaaaaa=1
aaab=baaa
bb=aaa
bab=aabaa
baab=aba

Same cardinality

6 unique, 67 total

Σ#PresentationDescriptionRelated
8425⟨a, b | aab=ba, bbb=1⟩Finite non-commutative monoid with 24 elements18 iso, 3 anti-iso
105009⟨a, b | aba=bb, aabbb=1⟩Finite non-Abelian group with 24 elements30 iso
106935⟨a, b | bb=aa, aaabab=1⟩Finite non-Abelian group with 24 elements2 iso
1113372⟨a, b | aaaab=1, bbbbbb=1⟩Isomorphic to ℤ248 iso
1119638⟨a, b | aba=a, aaaab=bb⟩Finite non-commutative monoid with 24 elements
1120252⟨a, b | aba=b, aaaa=bab⟩Finite non-commutative monoid with 24 elements

Isomorphic instances

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

10 total

Σ#PresentationMapping
1111520⟨a, b | ababa=b, baaab=1⟩φ(a) = a, φ(b) = b
1111526⟨a, b | ababa=b, bbaaa=1⟩φ(a) = a, φ(b) = b
1113479⟨a, b | aaabb=1, bababa=1⟩φ(a) = ab, φ(b) = b
1113622⟨a, b | aabba=1, ababab=1⟩φ(a) = ab, φ(b) = b
1113643⟨a, b | aabba=1, bababa=1⟩φ(a) = ab, φ(b) = b
1113783⟨a, b | abbba=1, ababab=1⟩φ(a) = b, φ(b) = ab
1114300⟨a, b | abba=b, aaabbb=1⟩φ(a) = ab, φ(b) = a
1114306⟨a, b | abba=b, aabbba=1⟩φ(a) = ab, φ(b) = a
1114311⟨a, b | abba=b, ababab=1⟩φ(a) = aabaa, φ(b) = a
1114320⟨a, b | abba=b, baaabb=1⟩φ(a) = ab, φ(b) = a