#13259 ⟨a, b | bab=aba, bbb=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.
The zero element z satisfies zy = yz = z for all y.
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.

1aba2abbab2a3abaab2ba2b2aa4ab2ab2a2a5
11aba2abbab2a3abaab2ba2b2aa4ab2ab2a2a5
aaa2aba3ba2abaab2a4a5b2a2a5ab2aa5a5a5a5
bbbab2ba2abab2aa2a5a5a5b2a2a3a5a5a4a5
a2a2a3ba2a4a5a5b2a2a5a5a5a5a5a5a5a5a5
abababaab2a5a5ab2aa3a5a5a5a5a4a5a5a5a5
bababa2abaa5b2a2a5a5a5a5a4a5a5a5a5a5a5
b2b2b2aa2b2a2a5a3ba2a5a5a5a4a5a5a5a5a5
a3a3a4a5a5a5a5a5a5a5a5a5a5a5a5a5a5
abaabaa5a5a5a5a5a5a5a5a5a5a5a5a5a5a5
ab2ab2ab2aa3a5a5a4a5a5a5a5a5a5a5a5a5a5
ba2ba2a5b2a2a5a5a5a4a5a5a5a5a5a5a5a5a5
b2ab2ab2a2a5a5a4a5a5a5a5a5a5a5a5a5a5a5
a4a4a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5
ab2aab2aa5a5a5a5a5a5a5a5a5a5a5a5a5a5a5
b2a2b2a2a5a4a5a5a5a5a5a5a5a5a5a5a5a5a5
a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5

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 ⇒ a5 [9]
2. ba3 ⇒ a5 [11]
3. aba2 ⇒ a5 [13]
4. a2b ⇒ ba2 [3]
5. bab ⇒ aba [1]
6. ab2a2 ⇒ a5 [12]
7. b3 ⇒ a2 [2]
# ab:bab=aba,bbb=aa a/b - -
aaaaaa=aaaaa
baaa=aaaaa
abaa=aaaaa
aab=baa
bab=aba
abbaa=aaaaa
bbb=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
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