#628 ⟨a, b | bb=aa, 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
aaa2ba5a3ba6a4ba7a5ba6baa7ba21ba3ba4
bbbaa2ba2a3ba3a4ba4a5ba5a6ba6a7ba71a
a2a2a3ba2a4ba3a5ba4a6ba5a7ba61ba7abba
bababa2a7ba31ba4aba5a2ba6a3ba7a4ba5a6
a3a3a4ba7a5ba6baa7ba21ba3aba4a2ba5ba6
ba2ba2ba3a4ba4a5ba5a6ba6a7ba71babaa2a3
a4a4a5ba4a6ba5a7ba61ba7aba2baa3ba2ba3
ba3ba3ba4aba5a2ba6a3ba7a4ba5baa6ba2a71
a5a5a6baa7ba21ba3aba4a2ba5a3ba6a4ba7b
ba4ba4ba5a6ba6a7ba71babaa2ba2a3ba3a4a5
a6a6a7ba61ba7aba2baa3ba2a4ba3a5ba4ba5
ba5ba5ba6a3ba7a4ba5baa6ba2a7ba31ba4aa2
a7a71ba3aba4a2ba5a3ba6a4ba7a5ba6baba2
ba6ba6ba71babaa2ba2a3ba3a4ba4a5ba5a6a7
ba7ba7ba5baa6ba2a7ba31ba4aba5a2ba6a3a4

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 [6]
2. ab ⇒ ba5 [5]
3. b2 ⇒ a2 [1]
# ab:bb=aa,abab=1 a/b - -
aaaaaaaa=1
ab=baaaaa
bb=aa

Same cardinality

20 unique, 117 total

Σ#PresentationPropertiesDescriptionφ
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
92051⟨a, b | aab=a, bbbb=b⟩Noc 164
103808⟨a, b | aaab=ba, abab=1⟩Grp 167
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.

58 total

Σ#PresentationMapping
81044⟨a, b | aa=1, ababbb=1⟩φ(a) = ba, φ(b) = a
81047⟨a, b | aa=1, abbbab=1⟩φ(a) = ba, φ(b) = a
81054⟨a, b | aa=1, bababb=1⟩φ(a) = ba, φ(b) = a
81130⟨a, b | aa=1, babbb=a⟩φ(a) = ba, φ(b) = a
104043⟨a, b | abb=aaa, abab=1⟩φ(a) = a, φ(b) = b
104048⟨a, b | abb=aaa, baba=1⟩φ(a) = a, φ(b) = b
104101⟨a, b | bab=aaa, aabb=1⟩φ(a) = baa, φ(b) = a
104103⟨a, b | bab=aaa, abba=1⟩φ(a) = baa, φ(b) = a
104105⟨a, b | bab=aaa, baab=1⟩φ(a) = baa, φ(b) = a
105867⟨a, b | aabb=1, aaaba=b⟩φ(a) = baa, φ(b) = a
105877⟨a, b | aabb=1, abaaa=b⟩φ(a) = baa, φ(b) = a
105899⟨a, b | abab=1, aaaba=b⟩φ(a) = a, φ(b) = b
105909⟨a, b | abab=1, abaaa=b⟩φ(a) = a, φ(b) = b
105931⟨a, b | abba=1, aaaba=b⟩φ(a) = baa, φ(b) = a
105960⟨a, b | abba=1, babbb=a⟩φ(a) = baa, φ(b) = a
105986⟨a, b | aaa=a, ababbb=1⟩φ(a) = ba, φ(b) = a
105989⟨a, b | aaa=a, abbbab=1⟩φ(a) = ba, φ(b) = a
105996⟨a, b | aaa=a, bababb=1⟩φ(a) = ba, φ(b) = a
106125⟨a, b | aab=b, ababbb=1⟩φ(a) = ba, φ(b) = a
106131⟨a, b | aab=b, abbbab=1⟩φ(a) = ba, φ(b) = a
106145⟨a, b | aab=b, bababb=1⟩φ(a) = ba, φ(b) = a
106148⟨a, b | aab=b, babbba=1⟩φ(a) = ba, φ(b) = a
106155⟨a, b | aab=b, bbabab=1⟩φ(a) = ba, φ(b) = a
106160⟨a, b | aab=b, bbbaba=1⟩φ(a) = ba, φ(b) = a
106216⟨a, b | aba=b, aabbbb=1⟩φ(a) = baaa, φ(b) = a
106224⟨a, b | aba=b, abbabb=1⟩φ(a) = baaa, φ(b) = a
106226⟨a, b | aba=b, abbbba=1⟩φ(a) = baaa, φ(b) = a
106231⟨a, b | aba=b, baabbb=1⟩φ(a) = baaa, φ(b) = a
106233⟨a, b | aba=b, babbab=1⟩φ(a) = baaa, φ(b) = a
106235⟨a, b | aba=b, bbaabb=1⟩φ(a) = baaa, φ(b) = a
109406⟨a, b | aa=1, aaababbb=1⟩φ(a) = ba, φ(b) = a
109412⟨a, b | aa=1, aaabbbab=1⟩φ(a) = ba, φ(b) = a
109424⟨a, b | aa=1, aabababb=1⟩φ(a) = ba, φ(b) = a
109427⟨a, b | aa=1, aababbba=1⟩φ(a) = ba, φ(b) = a
109432⟨a, b | aa=1, aabbabab=1⟩φ(a) = ba, φ(b) = a
109436⟨a, b | aa=1, aabbbaba=1⟩φ(a) = ba, φ(b) = a
109447⟨a, b | aa=1, abaaabbb=1⟩φ(a) = ba, φ(b) = a
109451⟨a, b | aa=1, abaabbab=1⟩φ(a) = ba, φ(b) = a
109455⟨a, b | aa=1, ababaabb=1⟩φ(a) = ba, φ(b) = a
109457⟨a, b | aa=1, abababba=1⟩φ(a) = ba, φ(b) = a
109459⟨a, b | aa=1, ababbaab=1⟩φ(a) = ba, φ(b) = a
109467⟨a, b | aa=1, abbaabab=1⟩φ(a) = ba, φ(b) = a
109475⟨a, b | aa=1, abbbaaab=1⟩φ(a) = ba, φ(b) = a
109489⟨a, b | aa=1, baaababb=1⟩φ(a) = ba, φ(b) = a
109493⟨a, b | aa=1, baababab=1⟩φ(a) = ba, φ(b) = a
109499⟨a, b | aa=1, babaaabb=1⟩φ(a) = ba, φ(b) = a
109700⟨a, b | aa=1, aababbb=a⟩φ(a) = ba, φ(b) = a
109710⟨a, b | aa=1, aabbbab=a⟩φ(a) = ba, φ(b) = a
109732⟨a, b | aa=1, abababb=a⟩φ(a) = ba, φ(b) = a
109736⟨a, b | aa=1, ababbba=a⟩φ(a) = ba, φ(b) = a
109744⟨a, b | aa=1, abbabab=a⟩φ(a) = ba, φ(b) = a
109766⟨a, b | aa=1, baaabbb=a⟩φ(a) = ba, φ(b) = a
109772⟨a, b | aa=1, baabbab=a⟩φ(a) = ba, φ(b) = a
109776⟨a, b | aa=1, babaabb=a⟩φ(a) = ba, φ(b) = a
1010005⟨a, b | aa=1, ababbb=aa⟩φ(a) = ba, φ(b) = a
1010017⟨a, b | aa=1, abbbab=aa⟩φ(a) = ba, φ(b) = a
1010043⟨a, b | aa=1, bababb=aa⟩φ(a) = ba, φ(b) = a
1010324⟨a, b | aa=1, babbb=aaa⟩φ(a) = ba, φ(b) = a