#12306 ⟨a, b | aaab=bb, abba=a⟩

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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.
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.

1aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
11aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
aaa2aba3a2ba4a3ba5a4ba6a5ba7a6baa7bab
bba4a3ba5a4ba6a5ba7a6baa7ba2aba3a2ba3b
a2a2a3a2ba4a3ba5a4ba6a5ba7a6baa7ba2aba2b
ababa5a4ba6a5ba7a6baa7ba2aba3a2ba4a3ba4b
a3a3a4a3ba5a4ba6a5ba7a6baa7ba2aba3a2ba3b
a2ba2ba6a5ba7a6baa7ba2aba3a2ba4a3ba5a4ba5b
a4a4a5a4ba6a5ba7a6baa7ba2aba3a2ba4a3ba4b
a3ba3ba7a6baa7ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5a5a6a5ba7a6baa7ba2aba3a2ba4a3ba5a4ba5b
a4ba4baa7ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6a6a7a6baa7ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5ba5ba2aba3a2ba4a3ba5a4ba6a5ba7a6baa7bab
a7a7aa7ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6ba6ba3a2ba4a3ba5a4ba6a5ba7a6baa7ba2aba2b
a7ba7ba4a3ba5a4ba6a5ba7a6baa7ba2aba3a2ba3b

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. a8 ⇒ a [8]
2. ba ⇒ a4 [9]
3. b2 ⇒ a3b [1]
# ab:aaab=bb,abba=a a/b - -
aaaaaaaa=a
ba=aaaa
bb=aaab

Same cardinality

20 unique, 169 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)
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.

6 total

Σ#PresentationMapping
1112314⟨a, b | aaab=bb, baba=a⟩φ(a) = a, φ(b) = b
1112318⟨a, b | aaab=bb, bbaa=a⟩φ(a) = a, φ(b) = b
1115765⟨a, b | aab=bb, abbba=a⟩φ(a) = aaaaa, φ(b) = b
1115781⟨a, b | aab=bb, babba=a⟩φ(a) = aaaaa, φ(b) = b
1115789⟨a, b | aab=bb, bbaba=a⟩φ(a) = aaaaa, φ(b) = b
1115793⟨a, b | aab=bb, bbbaa=a⟩φ(a) = aaaaa, φ(b) = b