#2051 ⟨a, b | aab=a, bbbb=b⟩

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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. Anti-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.

1aba2bab2a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
11aba2bab2a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
aaa2a3a3aa2aa2a3aa3aa2a2a3a
bbbab2ba2b2ab3ba3b2a2b3abb2a3b3a2bab3a3ba2ba3
a2a2a3aaa2a3a2a3aa2aa2a3a3aa2
bababa2ba3ba3baba2baba2ba3baba3baba2ba2ba3ba
b2b2b2ab3b2a2b3abb2a3b3a2bab2b3a3ba2b2aba3b2a2b2a3
a3a3aa2a2a3aa3aa2a3a2a3aaa2a3
ba2ba2ba3bababa2ba3ba2ba3baba2baba2ba3ba3baba2
b2ab2ab2a2b2a3b2a3b2ab2a2b2ab2a2b2a3b2ab2a3b2ab2a2b2a2b2a3b2a
b3b3b3abb3a2bab2b3a3ba2b2ab3ba3b2a2b3ab2a3b3a2b3a3
ba3ba3baba2ba2ba3baba3baba2ba3ba2ba3bababa2ba3
b2a2b2a2b2a3b2ab2ab2a2b2a3b2a2b2a3b2ab2a2b2ab2a2b2a3b2a3b2ab2a2
b3ab3ab3a2b3a3b3a3b3ab3a2b3ab3a2b3a3b3ab3a3b3ab3a2b3a2b3a3b3a
b2a3b2a3b2ab2a2b2a2b2a3b2ab2a3b2ab2a2b2a3b2a2b2a3b2ab2ab2a2b2a3
b3a2b3a2b3a3b3ab3ab3a2b3a3b3a2b3a3b3ab3a2b3ab3a2b3a3b3a3b3ab3a2
b3a3b3a3b3ab3a2b3a2b3a3b3ab3a3b3ab3a2b3a3b3a2b3a3b3ab3ab3a2b3a3

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. a4 ⇒ a [6]
2. ab ⇒ a3 [5]
3. b4 ⇒ b [2]
# ab:aab=a,bbbb=b a/b - -
aaaa=a
ab=aaa
bbbb=b

Same cardinality

20 unique, 171 total

Σ#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
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

Anti-isomorphic instances

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

4 total

Σ#PresentationMapping
104619⟨a, b | aaaa=a, aabb=b⟩φ(a) = bb, φ(b) = a
104621⟨a, b | aaaa=a, abab=b⟩φ(a) = bb, φ(b) = a
1114340⟨a, b | aaaa=a, aabab=b⟩φ(a) = b, φ(b) = aaa
1114346⟨a, b | aaaa=a, abaab=b⟩φ(a) = b, φ(b) = aaa