#16060 ⟨a, b | aaa=bb, abab=aa⟩

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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

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1aba2abbaa3ababa2baba4ba3(ba)2a5ba4ba5
11aba2abbaa3ababa2baba4ba3(ba)2a5ba4ba5
aaa2aba3ba5abaa4ba3ba3a2a5ba4a3a3ba5ba3
bbbaa3ba2baba4ba3(ba)2a5ba4ba4a3ba5ba5a4a5
a2a2a3ba5a4ba3ba3a5ba4ba4a3a3ba5a4a4ba3ba4
abababaa4ba3a2a5ba4a3a3ba5ba5a4ba3ba3a5a3
bababa2babba3a5(ba)2ba4a3a3ba2ba5a4ba3ba3a5a3
a3a3a4ba3a5ba4ba4a3ba5ba5a4a4ba3a5a5ba4ba5
abaababa3a2ba4a3a3ba5a4a4ba3ba3a5ba4ba4a3a4
ba2ba2ba3a5ba4a3a3ba5a4a4ba3ba3a5ba4ba4a3a4
babbab(ba)2ba4a3ba2ba5a4ba3ba3a5a5ba4a3a3ba5ba3
a4a4a5ba4a3ba5ba5a4ba3ba3a5a5ba4a3a3ba5ba3
ba3ba3ba4a3ba5a4a4ba3a5a5ba4ba4a3ba5ba5a4a5
(ba)2(ba)2a3ba2a4ba3ba3a5ba4ba4a3a3ba5a4a4ba3ba4
a5a5a3ba5a4ba3ba3a5ba4ba4a3a3ba5a4a4ba3ba4
ba4ba4ba5a4ba3a5a5ba4a3a3ba5ba5a4ba3ba3a5a3
ba5ba5ba3a5ba4a3a3ba5a4a4ba3ba3a5ba4ba4a3a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. a6 ⇒ a3 [8]
2. aba2 ⇒ ba3 [5]
3. a2b ⇒ ba5 [7]
4. b2 ⇒ a3 [1]
5. (ab)2 ⇒ a2 [2]
# ab:aaa=bb,abab=aa a/b - -
aaaaaa=aaa
abaa=baaa
aab=baaaaa
bb=aaa
abab=aa

Same cardinality

20 unique, 175 total

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