#1427 ⟨a, b | abba=b, baba=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

1ababb2ab2
11ababb2ab2
aa1abbab2b2
bbab2b2a1ab
ababb2ab21ab
b2b2ab1ab2ba
ab2ab2bab2ab1

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. b3 ⇒ 1 [10]
2. ba ⇒ ab2 [9]
3. a2 ⇒ 1 [12]
# ab:abba=b,baba=1 b/a - -
bbb=1
ba=abb
aa=1

Same cardinality

19 unique, 1859 total

Σ#PresentationPropertiesDescriptionφ
622⟨a, b | ab=aa, bb=1⟩Noc 6107
626⟨a, b | aaa=1, abb=1⟩Grp Com 6ℤ61373
7160⟨a, b | ab=aa, bb=b⟩Noc 64
7245⟨a, b | aa=a, bbb=a⟩Noc Com 6ℕ(6 = 3)49
7257⟨a, b | aa=b, bbb=a⟩Noc Com 6ℕ(6 = 1)61
7258⟨a, b | aa=b, bbb=b⟩Noc Com 6ℕ(6 = 2)55
8639⟨a, b | ab=aa, baa=b⟩Noc 613
8644⟨a, b | ab=aa, bbb=a⟩Noc Com 6ℕ(6 = 4)46
8893⟨a, b | aa=a, abbb=b⟩Noc 619
81011⟨a, b | ab=a, baa=bb⟩Noc 613
91581⟨a, b | aaa=aa, abb=b⟩Noc 65
91648⟨a, b | aab=bb, abb=a⟩Noc Com 613
91686⟨a, b | abb=aa, bbb=a⟩Noc Com 6ℕ(6 = 5)49
93075⟨a, b | ab=a, aaaa=bb⟩Noc Com 614
93132⟨a, b | ab=a, bbbb=aa⟩Noc Com 65
93134⟨a, b | ab=a, bbbb=ba⟩Noc 62
93193⟨a, b | ab=a, bbb=aaa⟩Noc Com 69
93199⟨a, b | ab=a, bbb=bba⟩Noc 63
1124145⟨a, b | aa=a, abbbbba=b⟩Noc Com 6

Isomorphic instances

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

66 total

Σ#PresentationMapping
91513⟨a, b | aab=ba, abab=1⟩φ(a) = b, φ(b) = a
91518⟨a, b | aab=ba, baba=1⟩φ(a) = b, φ(b) = a
91806⟨a, b | abab=1, aabba=1⟩φ(a) = b, φ(b) = a
91810⟨a, b | abab=1, abbaa=1⟩φ(a) = b, φ(b) = a
91811⟨a, b | abab=1, abbba=1⟩φ(a) = a, φ(b) = b
91815⟨a, b | abab=1, bbaaa=1⟩φ(a) = b, φ(b) = a
104889⟨a, b | aab=ba, aaabb=1⟩φ(a) = b, φ(b) = a
104892⟨a, b | aab=ba, aabba=1⟩φ(a) = b, φ(b) = a
104898⟨a, b | aab=ba, abbaa=1⟩φ(a) = b, φ(b) = a
104903⟨a, b | aab=ba, baaab=1⟩φ(a) = b, φ(b) = a
104910⟨a, b | aab=ba, bbaaa=1⟩φ(a) = b, φ(b) = a
1110701⟨a, b | aaab=aba, abab=1⟩φ(a) = b, φ(b) = a
1110706⟨a, b | aaab=aba, baba=1⟩φ(a) = b, φ(b) = a
1110836⟨a, b | aaba=baa, abab=1⟩φ(a) = b, φ(b) = a
1110841⟨a, b | aaba=baa, baba=1⟩φ(a) = b, φ(b) = a
1110916⟨a, b | aabb=aba, baba=1⟩φ(a) = a, φ(b) = b
1111020⟨a, b | abba=aab, baba=1⟩φ(a) = a, φ(b) = b
1111043⟨a, b | abba=baa, baba=1⟩φ(a) = a, φ(b) = b
1111048⟨a, b | abba=bab, baba=1⟩φ(a) = b, φ(b) = a
1111344⟨a, b | aabab=a, aabba=1⟩φ(a) = b, φ(b) = a
1111350⟨a, b | aabab=a, abbaa=1⟩φ(a) = b, φ(b) = a
1111352⟨a, b | aabab=a, abbba=1⟩φ(a) = a, φ(b) = b
1111355⟨a, b | aabab=a, baaab=1⟩φ(a) = b, φ(b) = a
1111357⟨a, b | aabab=a, baabb=1⟩φ(a) = a, φ(b) = b
1111362⟨a, b | aabab=a, bbaaa=1⟩φ(a) = b, φ(b) = a
1111363⟨a, b | aabab=a, bbaab=1⟩φ(a) = a, φ(b) = b
1111366⟨a, b | aabab=a, bbbaa=1⟩φ(a) = a, φ(b) = b
1111499⟨a, b | ababa=a, abbaa=1⟩φ(a) = b, φ(b) = a
1111501⟨a, b | ababa=a, abbba=1⟩φ(a) = a, φ(b) = b
1111504⟨a, b | ababa=a, baaab=1⟩φ(a) = b, φ(b) = a
1111506⟨a, b | ababa=a, baabb=1⟩φ(a) = a, φ(b) = b
1111572⟨a, b | abbba=b, baaab=1⟩φ(a) = b, φ(b) = a
1111579⟨a, b | abbba=b, bbaaa=1⟩φ(a) = b, φ(b) = a
1111586⟨a, b | baaab=a, baabb=1⟩φ(a) = a, φ(b) = b
1113455⟨a, b | aaabb=1, abaaba=1⟩φ(a) = b, φ(b) = a
1113473⟨a, b | aaabb=1, baabaa=1⟩φ(a) = b, φ(b) = a
1113610⟨a, b | aabba=1, aabaab=1⟩φ(a) = b, φ(b) = a
1113619⟨a, b | aabba=1, abaaba=1⟩φ(a) = b, φ(b) = a
1113637⟨a, b | aabba=1, baabaa=1⟩φ(a) = b, φ(b) = a
1113787⟨a, b | abbba=1, abbabb=1⟩φ(a) = a, φ(b) = b
1113796⟨a, b | abbba=1, babbab=1⟩φ(a) = a, φ(b) = b
1115146⟨a, b | aab=ba, aabaab=1⟩φ(a) = b, φ(b) = a
1115155⟨a, b | aab=ba, abaaba=1⟩φ(a) = b, φ(b) = a
1115173⟨a, b | aab=ba, baabaa=1⟩φ(a) = b, φ(b) = a
1116975⟨a, b | abab=1, aabbaab=1⟩φ(a) = a, φ(b) = b
1116985⟨a, b | abab=1, abaabba=1⟩φ(a) = a, φ(b) = b
1116993⟨a, b | abab=1, abbaaba=1⟩φ(a) = a, φ(b) = b
1116995⟨a, b | abab=1, abbabba=1⟩φ(a) = b, φ(b) = a
1117006⟨a, b | abab=1, baabbaa=1⟩φ(a) = a, φ(b) = b
1117013⟨a, b | abab=1, bbaabaa=1⟩φ(a) = a, φ(b) = b
1117502⟨a, b | abab=1, aabbaa=a⟩φ(a) = b, φ(b) = a
1117510⟨a, b | abab=1, abaaab=a⟩φ(a) = b, φ(b) = a
1117519⟨a, b | abab=1, abbaaa=a⟩φ(a) = b, φ(b) = a
1117521⟨a, b | abab=1, abbaab=a⟩φ(a) = a, φ(b) = b
1117524⟨a, b | abab=1, abbbaa=a⟩φ(a) = a, φ(b) = b
1117530⟨a, b | abab=1, baaaba=a⟩φ(a) = b, φ(b) = a
1117534⟨a, b | abab=1, baabba=a⟩φ(a) = a, φ(b) = b
1117539⟨a, b | abab=1, babbaa=b⟩φ(a) = b, φ(b) = a
1117540⟨a, b | abab=1, bbaaaa=a⟩φ(a) = b, φ(b) = a
1117543⟨a, b | abab=1, bbbaaa=a⟩φ(a) = a, φ(b) = b
1118033⟨a, b | abab=1, aabba=ab⟩φ(a) = a, φ(b) = b
1118040⟨a, b | abab=1, abaab=aa⟩φ(a) = b, φ(b) = a
1118042⟨a, b | abab=1, abaab=ba⟩φ(a) = a, φ(b) = b
1118050⟨a, b | abab=1, abbaa=ba⟩φ(a) = a, φ(b) = b
1118060⟨a, b | abab=1, baaba=aa⟩φ(a) = b, φ(b) = a
1118061⟨a, b | abab=1, baaba=ab⟩φ(a) = a, φ(b) = b