#6205 ⟨a, b | aba=b, aaaabb=1⟩

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Properties

Elements

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

Complete 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
  1. a8 ⇒ 1
  2. ab ⇒ ba7
  3. b2 ⇒ a4 
# ab:aba=b,aaaabb=1 a/b
aaaaaaaa=1
ab=baaaaaaa
bb=aaaa

Cayley table

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
aaa2ba7a3ba4baa5ba2a6ba3a7ba41ba5ba6
bbbaa4ba2a5ba3a6ba4a7ba51ba6aba7a2a3
a2a2a3ba6a4ba7a5ba6baa7ba21ba3aba4ba5
bababa2a3ba3a4ba4a5ba5a6ba6a7ba71baa2
a3a3a4ba5a5ba6a6ba7a7b1baaba2a2ba3ba4
ba2ba2ba3a2ba4a3ba5a4ba6a5ba7a6ba7ba1a
a4a4a5ba4a6ba5a7ba61ba7aba2baa3ba2ba3
ba3ba3ba4aba5a2ba6a3ba7a4ba5baa6ba2a71
a5a5a6ba3a7ba41ba5aba6a2ba7a3ba4baba2
ba4ba4ba51ba6aba7a2ba3baa4ba2a5ba3a6a7
a6a6a7ba21ba3aba4a2ba5a3ba6a4ba7a5bba
ba5ba5ba6a7ba71babaa2ba2a3ba3a4ba4a5a6
a7a71baaba2a2ba3a3ba4a4ba5a5ba6a6ba7b
ba6ba6ba7a6ba7ba1ba2aba3a2ba4a3ba5a4a5
ba7ba7ba5baa6ba2a7ba31ba4aba5a2ba6a3a4

Right Cayley graph

Left Cayley graph

Others with same cardinality

20 unique, 172 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
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
1116028⟨a, b | aaa=ab, babb=bb⟩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

Other isomorphic instances

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

3 total

Σ#PresentationMapping
106208⟨a, b | aba=b, aaabba=1⟩φ(a) = a, φ(b) = b
106213⟨a, b | aba=b, aabbaa=1⟩φ(a) = a, φ(b) = b
106228⟨a, b | aba=b, baaaab=1⟩φ(a) = a, φ(b) = b