#1331 ⟨a, b | aaaa=b, bbbb=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Staircase diagram
  4. Cayley table
  5. Right Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Elements

The order of x is the least n such that xn = 1.

Staircase diagram

Cayley table

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a151
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a151a
a3a3a4a5a6a7a8a9a10a11a12a13a14a151aa2
a4a4a5a6a7a8a9a10a11a12a13a14a151aa2a3
a5a5a6a7a8a9a10a11a12a13a14a151aa2a3a4
a6a6a7a8a9a10a11a12a13a14a151aa2a3a4a5
a7a7a8a9a10a11a12a13a14a151aa2a3a4a5a6
a8a8a9a10a11a12a13a14a151aa2a3a4a5a6a7
a9a9a10a11a12a13a14a151aa2a3a4a5a6a7a8
a10a10a11a12a13a14a151aa2a3a4a5a6a7a8a9
a11a11a12a13a14a151aa2a3a4a5a6a7a8a9a10
a12a12a13a14a151aa2a3a4a5a6a7a8a9a10a11
a13a13a14a151aa2a3a4a5a6a7a8a9a10a11a12
a14a14a151aa2a3a4a5a6a7a8a9a10a11a12a13
a15a151aa2a3a4a5a6a7a8a9a10a11a12a13a14

Right Cayley graph

Rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
#RuleProof
1. a16 ⇒ 1 [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,bbbb=1 a/b
aaaaaaaaaaaaaaaa=1
b=aaaa

Same cardinality

20 unique, 108 total

Σ#PresentationDescriptionRelated
8628⟨a, b | bb=aa, abab=1⟩Finite non-Abelian group with 16 elements58 iso
92051⟨a, b | aab=a, bbbb=b⟩Finite non-commutative monoid with 16 elements4 anti-iso
103808⟨a, b | aaab=ba, abab=1⟩Finite non-Abelian group with 16 elements7 iso
104630⟨a, b | aaaa=a, bbbb=a⟩Isomorphic to ℕ(16 = 4)1 iso
104648⟨a, b | aaaa=b, bbbb=a⟩Isomorphic to ℕ(16 = 1)5 iso
106205⟨a, b | aba=b, aaaabb=1⟩Finite non-Abelian group with 16 elements3 iso
1112164⟨a, b | aaaa=aa, bbbb=a⟩Isomorphic to ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Isomorphic to ℕ(16 = 5)2 iso
1112212⟨a, b | aaaa=bb, bbbb=a⟩Isomorphic to ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Finite non-commutative monoid with 16 elements6 iso
1113251⟨a, b | bab=aab, bbb=aa⟩Finite non-commutative monoid with 16 elements
1113259⟨a, b | bab=aba, bbb=aa⟩Finite non-commutative monoid with 16 elements
1116028⟨a, b | aaa=ab, babb=bb⟩Finite non-commutative monoid with 16 elements
1116032⟨a, b | aaa=ab, bbaa=bb⟩Finite non-commutative monoid with 16 elements
1116060⟨a, b | aaa=bb, abab=aa⟩Finite non-commutative monoid with 16 elements
1116371⟨a, b | aba=aa, bbbb=ab⟩Finite non-commutative monoid with 16 elements
1118811⟨a, b | aaa=b, abbbbb=b⟩Isomorphic to ℕ(16 = 3)2 iso
1120251⟨a, b | aba=b, aaaa=abb⟩Finite non-commutative monoid with 16 elements
1121039⟨a, b | ab=aa, bbba=bbb⟩Finite non-commutative monoid with 16 elements
1124660⟨a, b | aa=a, ababab=bb⟩Finite non-commutative monoid with 16 elements

Isomorphic instances

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

67 total

Σ#PresentationMapping
91713⟨a, b | aaaa=1, abbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
91717⟨a, b | aaaa=1, babbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
91718⟨a, b | aaaa=1, bbabb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
105796⟨a, b | aaab=1, bbbbb=a⟩φ(a) = aaaaa, φ(b) = a
105860⟨a, b | aaba=1, bbbbb=a⟩φ(a) = aaaaa, φ(b) = a
106027⟨a, b | aaa=b, abbbbb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
106034⟨a, b | aaa=b, babbbb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
106036⟨a, b | aaa=b, bbabbb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
1110669⟨a, b | aaaa=bbb, bbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1111108⟨a, b | aaaaa=a, abbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111112⟨a, b | aaaaa=a, babbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111113⟨a, b | aaaaa=a, bbabb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111177⟨a, b | aaaab=b, abbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111185⟨a, b | aaaab=b, babbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111189⟨a, b | aaaab=b, bbabb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111191⟨a, b | aaaab=b, bbbab=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111192⟨a, b | aaaab=b, bbbba=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111235⟨a, b | aaaba=b, abbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111243⟨a, b | aaaba=b, babbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111247⟨a, b | aaaba=b, bbabb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111249⟨a, b | aaaba=b, bbbab=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111250⟨a, b | aaaba=b, bbbba=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111335⟨a, b | aabaa=b, abbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111340⟨a, b | aabaa=b, babbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111342⟨a, b | aabaa=b, bbabb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111679⟨a, b | aaaa=bb, aabbb=1⟩φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
1111682⟨a, b | aaaa=bb, ababb=1⟩φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
1111683⟨a, b | aaaa=bb, abbab=1⟩φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
1111684⟨a, b | aaaa=bb, abbba=1⟩φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
1111687⟨a, b | aaaa=bb, baabb=1⟩φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
1111688⟨a, b | aaaa=bb, babab=1⟩φ(a) = aaa, φ(b) = aaaaaaaaaaaaaa
1114987⟨a, b | aaa=bb, aabbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1114995⟨a, b | aaa=bb, abbabb=1⟩φ(a) = aaaaaa, φ(b) = a
1114997⟨a, b | aaa=bb, abbbba=1⟩φ(a) = aaaaaa, φ(b) = a
1115002⟨a, b | aaa=bb, baabbb=1⟩φ(a) = aaaaaa, φ(b) = a
1115004⟨a, b | aaa=bb, babbab=1⟩φ(a) = aaaaaa, φ(b) = a
1115006⟨a, b | aaa=bb, bbaabb=1⟩φ(a) = aaaaaa, φ(b) = a
1116575⟨a, b | aaaa=1, aaabbbb=1⟩φ(a) = aaaa, φ(b) = a
1116585⟨a, b | aaaa=1, aabbabb=1⟩φ(a) = aaaa, φ(b) = a
1116588⟨a, b | aaaa=1, aabbbba=1⟩φ(a) = aaaa, φ(b) = a
1116598⟨a, b | aaaa=1, abababb=1⟩φ(a) = aaaa, φ(b) = a
1116599⟨a, b | aaaa=1, ababbab=1⟩φ(a) = aaaa, φ(b) = a
1116603⟨a, b | aaaa=1, abbaabb=1⟩φ(a) = aaaa, φ(b) = a
1116604⟨a, b | aaaa=1, abbabab=1⟩φ(a) = aaaa, φ(b) = a
1116605⟨a, b | aaaa=1, abbabba=1⟩φ(a) = aaaa, φ(b) = a
1116615⟨a, b | aaaa=1, baaabbb=1⟩φ(a) = aaaa, φ(b) = a
1116618⟨a, b | aaaa=1, baabbab=1⟩φ(a) = aaaa, φ(b) = a
1116621⟨a, b | aaaa=1, bababab=1⟩φ(a) = aaaa, φ(b) = a
1116626⟨a, b | aaaa=1, bbaaabb=1⟩φ(a) = aaaa, φ(b) = a
1117116⟨a, b | aaaa=1, aabbbb=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117128⟨a, b | aaaa=1, ababbb=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117132⟨a, b | aaaa=1, abbabb=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117134⟨a, b | aaaa=1, abbbab=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117136⟨a, b | aaaa=1, abbbba=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117150⟨a, b | aaaa=1, babbab=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117154⟨a, b | aaaa=1, bbaabb=a⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1117664⟨a, b | aaaa=1, abbbb=aa⟩φ(a) = aaaa, φ(b) = a
1117682⟨a, b | aaaa=1, bbabb=aa⟩φ(a) = aaaa, φ(b) = a
1121511⟨a, b | aab=1, bbbbbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaaaa
1121647⟨a, b | aba=1, bbbbbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaaaa
1123787⟨a, b | aa=b, bbbbbbbb=1⟩φ(a) = a, φ(b) = aa
1126111⟨a, b | aa=1, abbbbbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1126163⟨a, b | aa=1, babbbbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1126177⟨a, b | aa=1, bbabbbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1126181⟨a, b | aa=1, bbbabbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1126182⟨a, b | aa=1, bbbbabbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1126710⟨a, b | aa=1, bbbbbbbb=a⟩φ(a) = aaaaaaaa, φ(b) = a