#13251 ⟨a, b | bab=aab, 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.

1ababbab2abaab2b2ab3ab2aab3b4b5b6b7
11ababbab2abaab2b2ab3ab2aab3b4b5b6b7
aab3abb4abaab2b6b5ab2aab3b7b6b6b7b7b7
bbbab2b4b2ab3b6b5ab3b4b7b6b5b6b7b7
abababaab2b6ab2aab3b7b7b6b6b7b7b7b7b7b7
babab4b4b5b6b5b7b6b7b6b7b7b7b7b7b7
b2b2b2ab3b5ab3b4b7b6b6b5b7b7b6b7b7b7
abaabab6b6b7b7b7b7b7b7b7b7b7b7b7b7b7
ab2ab2ab2aab3b7b6b6b7b7b7b7b7b7b7b7b7b7
b2ab2ab5b5b6b7b6b7b7b7b7b7b7b7b7b7b7
b3b3ab3b4b6b6b5b7b7b7b6b7b7b7b7b7b7
ab2aab2ab7b7b7b7b7b7b7b7b7b7b7b7b7b7b7
ab3ab3b6b6b7b7b7b7b7b7b7b7b7b7b7b7b7
b4b4b6b5b7b7b6b7b7b7b7b7b7b7b7b7b7
b5b5b7b6b7b7b7b7b7b7b7b7b7b7b7b7b7
b6b6b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7
b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7

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. b8 ⇒ b7 [6]
2. ab4 ⇒ b6 [5]
3. bab ⇒ b4 [3]
4. b3a ⇒ ab3 [4]
5. a2 ⇒ b3 [2]
# ab:bab=aab,bbb=aa b/a - -
bbbbbbbb=bbbbbbb
abbbb=bbbbbb
bab=bbbb
bbba=abbb
aa=bbb

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
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