#16028 ⟨a, b | aaa=ab, babb=bb⟩

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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

Properties

Elements

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.

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
aaa2a3a3a4a4a5a5a6a6a7a7a5a5a6a7
bbbaba5ba2ba6ba3ba7ba4ba5ba5ba6ba6ba7ba7ba5ba6
a2a2a3a4a4a5a5a6a6a7a7a5a5a6a6a7a5
bababa2ba3ba3ba4ba4ba5ba5ba6ba6ba7ba7ba5ba5ba6ba7
a3a3a4a5a5a6a6a7a7a5a5a6a6a7a7a5a6
ba2ba2ba3ba4ba4ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba5
a4a4a5a6a6a7a7a5a5a6a6a7a7a5a5a6a7
ba3ba3ba4ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba6
a5a5a6a7a7a5a5a6a6a7a7a5a5a6a6a7a5
ba4ba4ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba7
a6a6a7a5a5a6a6a7a7a5a5a6a6a7a7a5a6
ba5ba5ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba5
a7a7a5a6a6a7a7a5a5a6a6a7a7a5a5a6a7
ba6ba6ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba6
ba7ba7ba5ba6ba6ba7ba7ba5ba5ba6ba6ba7ba7ba5ba5ba6ba7

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. a8 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b2 ⇒ ba5 [2]
# ab:aaa=ab,babb=bb a/b
aaaaaaaa=aaaaa
ab=aaa
bb=baaaaa

Same cardinality

20 unique, 175 total

Σ#PresentationDescriptionRelated
8628⟨a, b | bb=aa, abab=1⟩Finite non-Abelian group with 16 elements58 iso
91331⟨a, b | aaaa=b, bbbb=1⟩Isomorphic to ℤ1667 iso
92051⟨a, b | aab=a, bbbb=b⟩Finite non-commutative monoid with 16 elements4 anti-iso
103808⟨a, b | aaab=ba, abab=1⟩Finite non-Abelian group with 16 elements7 iso
104630⟨a, b | aaaa=a, bbbb=a⟩Isomorphic to ℕ(16 = 4)1 iso
104648⟨a, b | aaaa=b, bbbb=a⟩Isomorphic to ℕ(16 = 1)5 iso
106205⟨a, b | aba=b, aaaabb=1⟩Finite non-Abelian group with 16 elements3 iso
1112164⟨a, b | aaaa=aa, bbbb=a⟩Isomorphic to ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Isomorphic to ℕ(16 = 5)2 iso
1112212⟨a, b | aaaa=bb, bbbb=a⟩Isomorphic to ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Finite non-commutative monoid with 16 elements6 iso
1113251⟨a, b | bab=aab, bbb=aa⟩Finite non-commutative monoid with 16 elements
1113259⟨a, b | bab=aba, bbb=aa⟩Finite non-commutative monoid with 16 elements
1116032⟨a, b | aaa=ab, bbaa=bb⟩Finite non-commutative monoid with 16 elements
1116060⟨a, b | aaa=bb, abab=aa⟩Finite non-commutative monoid with 16 elements
1116371⟨a, b | aba=aa, bbbb=ab⟩Finite non-commutative monoid with 16 elements
1118811⟨a, b | aaa=b, abbbbb=b⟩Isomorphic to ℕ(16 = 3)2 iso
1120251⟨a, b | aba=b, aaaa=abb⟩Finite non-commutative monoid with 16 elements
1121039⟨a, b | ab=aa, bbba=bbb⟩Finite non-commutative monoid with 16 elements
1124660⟨a, b | aa=a, ababab=bb⟩Finite non-commutative monoid with 16 elements