#20251 ⟨a, b | aba=b, aaaa=abb⟩

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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.

1aba2baa3ba2a4ba3a5ba4a6ba5a7a8a9
11aba2baa3ba2a4ba3a5ba4a6ba5a7a8a9
aaa2ba5a3ba4baa5ba2a6ba3a7ba4a8a9a4
bbbaa9ba2a4ba3a5ba4a6ba5a7ba8baba2ba3
a2a2a3ba4a4ba5a5ba6baa7ba2a8ba3a9a4a5
bababa2a8ba3a9ba4a4ba5a5ba6baa7ba2ba3ba4
a3a3a4ba3a5ba4a6ba5a7ba8baa9ba2a4a5a6
ba2ba2ba3a7ba4a8ba5a9ba4baa5ba2a6ba3ba4ba5
a4a4a5ba2a6ba3a7ba4a8ba5a9ba4baa5a6a7
ba3ba3ba4a6ba5a7ba8baa9ba2a4ba3a5ba4ba5b
a5a5a6baa7ba2a8ba3a9ba4a4ba5a5ba6a7a8
ba4ba4ba5a5ba6baa7ba2a8ba3a9ba4a4ba5bba
a6a6a7ba8baa9ba2a4ba3a5ba4a6ba5a7a8a9
ba5ba5ba4baa5ba2a6ba3a7ba4a8ba5a9bbaba2
a7a7a8ba5a9ba4baa5ba2a6ba3a7ba4a8a9a4
a8a8a9ba4a4ba5a5ba6baa7ba2a8ba3a9a4a5
a9a9a4ba3a5ba4a6ba5a7ba8baa9ba2a4a5a6

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. a10 ⇒ a4 [10]
2. ba6 ⇒ b [8]
3. ab ⇒ ba5 [7]
4. b2 ⇒ a9 [9]
# ab:aba=b,aaaa=abb a/b - -
aaaaaaaaaa=aaaa
baaaaaa=b
ab=baaaaa
bb=aaaaaaaaa

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
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
1121039⟨a, b | ab=aa, bbba=bbb⟩Noc 16
1124660⟨a, b | aa=a, ababab=bb⟩Noc 16