#3808 ⟨a, b | aaab=ba, abab=1⟩

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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
  8. Isomorphic instances

Properties

Finite structure

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

Cayley table

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
aaa2ba3a3ba4a4ba5a5ba6a6ba7a7b1baba2
bbbaa4ba2a5ba3a6ba4a7ba51ba6aba7a2a3
a2a2a3ba6a4ba7a5ba6baa7ba21ba3aba4ba5
bababa2a7ba31ba4aba5a2ba6a3ba7a4ba5a6
a3a3a4baa5ba2a6ba3a7ba41ba5aba6a2ba7b
ba2ba2ba3a2ba4a3ba5a4ba6a5ba7a6ba7ba1a
a4a4a5ba4a6ba5a7ba61ba7aba2baa3ba2ba3
ba3ba3ba4a5ba5a6ba6a7ba71babaa2ba2a3a4
a5a5a6ba7a7b1baaba2a2ba3a3ba4a4ba5ba6
ba4ba4ba51ba6aba7a2ba3baa4ba2a5ba3a6a7
a6a6a7ba21ba3aba4a2ba5a3ba6a4ba7a5bba
ba5ba5ba6a3ba7a4ba5baa6ba2a7ba31ba4aa2
a7a71ba5aba6a2ba7a3ba4baa5ba2a6ba3ba4
ba6ba6ba7a6ba7ba1ba2aba3a2ba4a3ba5a4a5
ba7ba7babaa2ba2a3ba3a4ba4a5ba5a6ba6a71

Right Cayley graph

Left Cayley graph

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 ⇒ 1 [8]
2. ab ⇒ ba3 [9]
3. b2 ⇒ a4 [10]
# ab:aaab=ba,abab=1 a/b - -
aaaaaaaa=1
ab=baaa
bb=aaaa

Same cardinality

20 unique, 168 total

Σ#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
92051⟨a, b | aab=a, bbbb=b⟩Noc 164
104630⟨a, b | aaaa=a, bbbb=a⟩Noc Com 16ℕ(16 = 4)1
104648⟨a, b | aaaa=b, bbbb=a⟩Noc Com 16ℕ(16 = 1)5
106205⟨a, b | aba=b, aaaabb=1⟩Grp 163
1112164⟨a, b | aaaa=aa, bbbb=a⟩Noc Com 16ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Noc Com 16ℕ(16 = 5)2
1112212⟨a, b | aaaa=bb, bbbb=a⟩Noc Com 16ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Noc 166
1113251⟨a, b | bab=aab, bbb=aa⟩Noc 16
1113259⟨a, b | bab=aba, bbb=aa⟩Noc 16
1116028⟨a, b | aaa=ab, babb=bb⟩Noc 16
1116032⟨a, b | aaa=ab, bbaa=bb⟩Noc 16
1116060⟨a, b | aaa=bb, abab=aa⟩Noc 16
1116371⟨a, b | aba=aa, bbbb=ab⟩Noc 16
1118811⟨a, b | aaa=b, abbbbb=b⟩Noc Com 16ℕ(16 = 3)2
1120251⟨a, b | aba=b, aaaa=abb⟩Noc 16
1121039⟨a, b | ab=aa, bbba=bbb⟩Noc 16
1124660⟨a, b | aa=a, ababab=bb⟩Noc 16

Isomorphic instances

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

7 total

Σ#PresentationMapping
103813⟨a, b | aaab=ba, baba=1⟩φ(a) = a, φ(b) = b
103974⟨a, b | abba=bb, baba=1⟩φ(a) = b, φ(b) = a
105628⟨a, b | abab=1, aaabba=1⟩φ(a) = a, φ(b) = b
105634⟨a, b | abab=1, aabbaa=1⟩φ(a) = a, φ(b) = b
105643⟨a, b | abab=1, abbaaa=1⟩φ(a) = a, φ(b) = b
105647⟨a, b | abab=1, abbbba=1⟩φ(a) = b, φ(b) = a
105655⟨a, b | abab=1, bbaaaa=1⟩φ(a) = a, φ(b) = b