#11515 ⟨a, b | ababa=b, abbaa=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

1aba2abbab2a2babaab2babb3a2baa2b2(ab)2ab3bab2a(ab)2a2b3abab2bab3a2bab2abab3a2bab3
11aba2abbab2a2babaab2babb3a2baa2b2(ab)2ab3bab2a(ab)2a2b3abab2bab3a2bab2abab3a2bab3
aaa2abb2a2babaab2b3a2baa2b2(ab)2ab3bab21a(ab)2a2b3abab2bab3ba2bab2abab3baa2bab3bab
bbbab2abab3babab2b3abaa2b3bab2ab31a2bab3(ab)2a2bab3aa2baabab2a2baba(ab)2a2b2a2bab2
a2a2b2a2bab2b3a2baa2b2ab3bab21a(ab)2a2b3abab2abab3ba2bab2abab3abbaa2bab3ababab(ab)2
abababaab2a2bab3(ab)2a2b2ab3a2bababab2a2b3ababa(ab)2b2abab3a2bab2a2bab2b3a2bbab31ba
babaabab3babb3abaa2b3bab21a2bab3(ab)2a2bab3aba2baabab2a2babb2a(ab)2a2b2ab2a2bab2ab3
b2b2ab2b3a2b2ab3bab21a2b3abab2abab3ba2bab2a2abab3abbaa2bab3a2babababa2ba(ab)2a(ab)2
a2ba2ba2baa2b2baba(ab)21a2b3bab2aba2bab2ba2(ab)2bab3ab2a2bab3b2abab2baab3b3abab3aaba
abaabaa2bab3(ab)2ab3a2bababab2ababa(ab)2b2abab3a2abbab2a2bab2b3a2bab2bab31a2b2baa2b3
ab2ab2a2b2ab31a2b3abab2aba2bab2a2abab3abbab2a2bab3a2babababb3a2ba(ab)2bab2a(ab)2bab3
babbaba2b3bab2a2bab2a2(ab)2bab3a2bab3b2a2babab2baab3a2bab3a2b2abab3aa(ab)21abaabbab2
b3b3bab21(ab)2bab3ababab2a2bbaabb2a(ab)2abab3a2b2babab2a2bab2abaa2b3ab3a2bab3a2a2ba
a2baa2bababa(ab)2a2b3bab2aba2bab2a2(ab)2bab3ab2a2bab3b2a2babab2baab3b3a2b2abab3a1abab
a2b2a2b21a2b3aba2bab2a2abbab2a2bab3a2babaab2babb3a2ba(ab)2ab3bab2a(ab)2abab2bab3abab3
(ab)2(ab)2babab2bab2a(ab)2abab3babab2b3a2bab2abaa2b3bab2ab31a2bab3a2bab3aa2baa2baba2b2
ab3ab3abab2aa(ab)2abab3a2aba2bab2b3abaa2bab2bab3a2bab31(ab)2a2b2baa2baba2b3babb2bab2
bab2bab2(ab)2bab3babab2a2bbab2a(ab)2abab3a2b2babab2b3a2bab2abaa2b3ab31a2bab3a2aa2baab
a(ab)2a(ab)2aba2bab2abaab2bab3a2bab3(ab)2a2b2ab3baa2bababab2a2b3ababb2abab3a2bab2b3a2b1
a2b3a2b3a2bab2a2bab3a2bab3b2a2bbaab3a2bab3a2b2abab3babaa(ab)21ababab2abb(ab)2ab2abab2
abab2abab2a(ab)2abab3aba2bab2b3abaab2bab3a2bab31(ab)2a2b2ab3baa2baba2b3ababb2a2bab2a2b
bab3bab3a2bbaa2baa2b2abab3baba(ab)21a2b3ababab2aba2bab2ba2(ab)2ab2a2bab3b2abab2ab3b3a
a2bab2a2bab2bab3a2bab3a2bbaab3a2baa2b2abab3babaa(ab)21a2b3ababab2abba2(ab)2ab2b2abab2b3
abab3abab3b3ababab21a2bab3(ab)2bab3aba2baabab2a2bbaabb2a(ab)2a2b2babab2a2bab2a2b3ab3a2
a2bab3a2bab3ab3a2baabab2ababa(ab)2abab3a2abbab2a2bab2b3abaa2bab2bab31(ab)2a2b2baba2b3b2

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. b4 ⇒ 1 [15]
2. b2a ⇒ ab2 [3]
3. ba2 ⇒ abab3 [13]
4. (ba)2 ⇒ a2b3 [5]
5. a3 ⇒ b2 [14]
# ab:ababa=b,abbaa=1 b/a - -
bbbb=1
bba=abb
baa=ababbb
baba=aabbb
aaa=bb

Same cardinality

6 unique, 67 total

Σ#PresentationPropertiesDescriptionφ
8425⟨a, b | aab=ba, bbb=1⟩Noc 2421
105009⟨a, b | aba=bb, aabbb=1⟩Grp 2430
106935⟨a, b | bb=aa, aaabab=1⟩Grp 242
1113372⟨a, b | aaaab=1, bbbbbb=1⟩Grp Com 24ℤ248
1119638⟨a, b | aba=a, aaaab=bb⟩Noc 24
1120252⟨a, b | aba=b, aaaa=bab⟩Noc 24

Isomorphic instances

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

10 total

Σ#PresentationMapping
1111520⟨a, b | ababa=b, baaab=1⟩φ(a) = a, φ(b) = b
1111526⟨a, b | ababa=b, bbaaa=1⟩φ(a) = a, φ(b) = b
1113479⟨a, b | aaabb=1, bababa=1⟩φ(a) = ab, φ(b) = b
1113622⟨a, b | aabba=1, ababab=1⟩φ(a) = ab, φ(b) = b
1113643⟨a, b | aabba=1, bababa=1⟩φ(a) = ab, φ(b) = b
1113783⟨a, b | abbba=1, ababab=1⟩φ(a) = b, φ(b) = ab
1114300⟨a, b | abba=b, aaabbb=1⟩φ(a) = ab, φ(b) = a
1114306⟨a, b | abba=b, aabbba=1⟩φ(a) = ab, φ(b) = a
1114311⟨a, b | abba=b, ababab=1⟩φ(a) = bab, φ(b) = a
1114320⟨a, b | abba=b, baaabb=1⟩φ(a) = ab, φ(b) = a