#16966 ⟨a, b | abab=1, aaabbba=1⟩

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Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. 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.

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 [23]
2. a4b ⇒ ba4 [15]
3. b2 ⇒ aba5 [24]
4. bab ⇒ a7 [25]
5. ba2b ⇒ a3ba7 [26]
6. aba3b ⇒ (ba3)2 [18]
# ab:abab=1,aaabbba=1 a/b - -
aaaaaaaa=1
aaaab=baaaa
bb=abaaaaa
bab=aaaaaaa
baab=aaabaaaaaaa
abaaab=baaabaaa

Same cardinality

4 unique, 12 total

Σ#PresentationPropertiesφ
103771⟨a, b | aaaa=bb, abab=1⟩Grp 485
1114993⟨a, b | aaa=bb, ababbb=1⟩Grp 482
1117503⟨a, b | abab=1, aabbaa=b⟩Grp 48
1121751⟨a, b | aaa=1, abbbbbb=b⟩Noc 481

Isomorphic instances

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

6 total

Σ#PresentationMapping
1116977⟨a, b | abab=1, aabbbaa=1⟩φ(a) = a, φ(b) = b
1116979⟨a, b | abab=1, aabbbba=1⟩φ(a) = b, φ(b) = a
1116996⟨a, b | abab=1, abbbaaa=1⟩φ(a) = a, φ(b) = b
1116998⟨a, b | abab=1, abbbbaa=1⟩φ(a) = b, φ(b) = a
1117015⟨a, b | abab=1, bbbaaaa=1⟩φ(a) = a, φ(b) = b
1118069⟨a, b | abab=1, bbaaa=ab⟩φ(a) = a, φ(b) = b