#1331 ⟨a, b | aaaa=b, bbbb=1⟩

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Contents

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

Properties

Commutative structure

Finite structure

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:
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. 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#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
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.

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